Nonstandard Analysis for Some Convergence Sequences Theories Via a Transfer Principle

Email: jawahirhussien2012@gmail.com Abstract: The purpose of this article to introduce nonstandard models for a familiar types of convergent sequences theories through a transfer principle. The nonstandard analysis principle discuss, that any statements on areal system can be extended to a similar structure over larger hyperreal system therefore the results that hold true on the original system, remains true in a hyperreal system if and only if its ∗-transform is true. We apply such technique to transfer a classical proof for real sequences theorems, therefore we obtain an equivalents nonstandard proof on hyperreal system which is often clear, shorter and uncoblicated.


Introduction
Nonstandard analysis is anew mathematical technique that widely use in dever field of mathematics and other science such as Statistics and economics (Goldblatt, 1998). It become a powerful, mathematical tool in the 20th century for providing a new method in order to formulating statements and proving theorems which yelding for an enlarge view of the mathematical land scape (Goldbring, 2014;Goldblatt, 2012).
The first who has been succeeded to demonestrated a rigorous foundation for the use of infinitesimal and infinite numbers in analysis (Lengyel, 1996), and provided a basic concepts of nonstandard analysis was Abraham Robinson during 1960 s. Therefore a new large field of numbers known as a hyperreal system includes areal number system, infinite, infinitesimals numbers which are non-zero, infinitely larg and small numbers were constructed (Keisler, 2000). In fact a hyperreal numbers system can be regarded as extension order field of areal numbers * areal number system infinitesimals. Moreovere A. Robinson has been showed that a relational structure over real numbers can be transferring to equivalent structure over hyperreal system (Hurd and Loeb, 1985). Therefore every statement holds true within real system remain true in hyperreal system hence this propriety for transferring statements known as a transfer principle which based on Robinsons approach (a nonstandard analysis).
There were a number of studies have examined a results on nonstandard analysis and its applications. Sun (2015) were introduced an applications economics, Similary (Duanmu, 2018) were applied the nonstandard analysis to Markov processes and Statistical Decision theory. An interesting application of a transfer principle for continuations of real functions to Levi-Civita field has been presented by (Bottazzi, 2018). A new approach to nonstandard analysis has been presented by (Abdeljalil, 2018), he proposed a very simple method in practice to nonstandard analysis without using the ultrafilter, (Ciurea, 2018) has constructed an approach for nonstandard analysis in a complete metric spaces.
We apply nonstandard analysis concepts to give nonstandard models equivalents for the classical theories of real sequences that converges. The method we used is a cording to Abraham Robinson for transferring properties from areal number system to ahyperreal system * via a transfer principle therefore all properties of sequences of the real numbers were preserved. The equivalent proofs we obtained often simpler and directly which is desire result.
The study compose of five section we organized it as follows:

The Structure of the Hyperreal Numbers System
The hyperreal number can be constructed within two approach, the first one as an extension order field of a set of real numbers, and the other as ultra-power. The two approaches allows us to extend arbitrary functions and relations from to * .

The Structure of the Hyperreal Number System as Complete Order Field
A real numbers system is a complete order field   , ,.,0  where +,  and < are the usual algebraic operation (relations of addition, multiplication and linear ordering) on . Recall that real system can be extend to an orderd field denoted by * which contains an isometric copy of but strictly larger called a hyperreal or nonstandard system (Staunton, 2013;Hurd and Loeb, 1985). Furthermore the construction of hyperreal number is reminiscent of the construction of the reals from the rationales numbers by means of equivalence classes of Cauchy sequence (Hurd and Loeb, 1985) (Staunton, 2013). In fact an elements of hyeperreal number * should be viewed as infinite sequence of real numbers. For example, the sequence 1,2,3…… should represent some infinite element of . However, many different sequences of rational numbers represent the same real number . Now we will introduce a basic arithmetical stricter of a hyper real and its relation to areal numbers.

Number Systems and Infinitesimal
An infinitesimal is known as a number that is smaller in magnitude than any non-zero real number and is larger than every negative real number or equivalently in absolute value it is smaller than 0 1 m for all m0 (Ponstein, 1975). Although zero is only infinitesimal number which belong to real number system, it may be extended in some way in order to include all infinitesimals (Ponstein, 1975). The following proposition illustrates some facts in a hyper areal numbers .
A non zero number k on is said to be: A hyperreal numbers b is said to be positive infinitesimal if b > 0 but less than a number o < a , negative infinitesimal if b < 0 but its greater than a negative number o < a . A hyperreal numbers is said to be infinitesimal if it either positive or negative infinitesimal or zero (Goldblatt, 2012;Keisler, 1976). Hens the sets of all infinitesimals numbers to which zero belong and the set of all hyperlarg numbers which containing a classical numbers all together are constitute a nonstandard numbers therefore they are clearly an extension of the real numbers .
Hens the large system of numbers which contains areal numbers system, infinite, infinitesimals numbers that are non-zero, known as nonstandard or hyperreal numbers and denoted by * .

Proposition. 2.3.6
A number b is negative infinitesimal if s < b < 0 for all 0 > s .

Proposition. 2.3.7
A number b is appreciable s < |b| < t for some s, t  . In facts all hyperreal numbers and an infinitesimals are (finite).

Proposition. 2.3.1
The product of an infinitesimal and a finite number is infinitesimal.
An integer m is limited exactly when it is standard and if m is nonstandard it said to be illimited, for The terms finite and infinite are often used for limited and unlimited. Notice that b is limited for some n  , unlimited iff for all, |b|<n, n  , appreciable if only if 1 n < |b|<n for some n  .
We now will introduce some basic a arithmetic operations on the hyper real numbers.

An Operations on the Hyperreal Numbers
Suppose a, b be infinitesimal, c, d be appreciable and e, f be unlimited numbers then the following operation are holds.

Addition
Observe that the limited numbers and infinitesimals are each a subring of * .

The Standard part of Hyperreal Numbers
The standard part play a basic role in infinitesimal theorem, it connect between the finite numbers of nonstandard analysis and the classical numbers i.e., it rounds off each finite hyperreal to the nearest real number therefore infinite hyperreal numbers never possess standard part.

Definition. 2.5.1
Let s  be finite (or a limited number) then the unique real number y that is infinitesimally close to s, (s  t) is called the standard part (or shadow) of x. we denote it by st(s) or sh(s). Thus we have t  sh(s) or (t  sh(sx)).

Theorem. 2.5.1
Every limited hyperreal numbers s is infinitely closed to exactly one real number called the standard part (shadow) of s.

Closeness of Hyperreal Numbers
Definition. 2.6.1 A number x and y in * are said to be near or infinitesimal close, if their difference x-y  is infinitesimal, thus x is infinitesimal if and only if x  y. We said that x, y are finitely close if x-y is finite and it written as x  y.
Every finite hyperreal number is infinitely close to exactly one real number, therefore existing of standard part of any infinite numbers depend on infinitely closeness to a finite.

The Monad and Galaxy Set of Hyperreal Number
For every hyperreal number, there exist two nonempty sets namely monad and galaxy which play an important role in infinitesimals theory.
In elementary Calculus, the pictorial device of an infinitesimal is used to illustrate part of a monad and an infinite telescope is used to illustrate part of an infinite galaxy (Stroyan and Luxemburg, 1977;Keisler, 2000;1976).
The galaxy of a set x is the set: The monad(0) is the set of infinitesimals and gala(0) is the set of finite hyperreal numbers. II. Any two monads are equal or disjoint. III. Let be two monads m(x) and m(y) be two monads then they are ether: The relation  is an equivalence relation on * .

Proof
The relation  equivalence relation if it satisfies the following condition.
For any finite hyperreal x, y, z  , we have: I.
x-x  0 is in * . II. x-y  y-x, so x  y implies y  x III. if x-y and y-z, in * then x-z is * Thus from I, II, III then () is an equivalence relation on * .

Theorem 2.7.1
The set monad (0) of infinitesimal elements is a subring of * and an ideal in galaxy (0). That is: (i). Sums, differences and products of infinitesimals are infinitesimal (ii). The product of an infinitesimal and a finite element is infinitesimal (For proof the see theorem (1.4) (Keisler, 1976)) Proposition 2.7.4 I. Two galaxies G(x) and G(y) are either equal if x-y is finite or disjoint II. If x  0 then m(x) is a translate of m(0) Therefore for every x   then: The quotient field G(0)/m(0) is isomorphic to the standard field of  (Andres and Rayo, 2015).
Proof m(0) is the kernel of the linear (over ) map st, i.e.: The following theorem explain existence and uniqueness prosperity of standard part.

Theorem. 2.8.1. (Standard Part Principle)
Every finite number x  is infinitely close to a unique real number y . Therefore every finite monad contains a uniquely number on .

Proof
Let x be in  is infinitely close to a unique real number y . Then every finite hyperreal number x is infinitely close to a unique real number.

Uniqueness
Consider y, z  and y  x, z  x. Since  is an equivalence relation we have y  z, hence y-z  0. But y-z is in , so y-z = 0 and y = z.

Existences
Let E = {y : y < x} be a nonempty set. A set E has an upper bound if there is real number y > 0 such that: |x| < y, whence -y < x < y so, -y  E hence y is an upper bound of E. Since is complete ordered field, so the set E has at least upper bound h. For every y  where y > 0 we have: , (Keisler, 1976).

Definition. 2.8.2. (The Standard Part Map.)
The map st: xst(x) called the standard part map. Clearly maps x onto st(x) since st(x) = x, when x  , furthermore it preserves algebraic structure as in the following theorem (Andres and Rayo, 2015).

Theorem 2.8.3
For every x, y  : The existence of standard of limited numbers follows from the Dedekind completeness of areal numbers . In fact the existence of standard part is a tentative formulation of completeness.

Theorem. 2.8.4
Every limited hyperreal x  is infinitely close to exactly one real number implies the completeness of (Goldblatt, 2012)[ theorem ( 5.8.1)].
In the following section is we will show the constructing of a hyperreal number system as linearly ordered field based on the ultra powers construction of nonstandard model.

A Construction of Hyper Real Number System as Ultra-Power
The construction of hyperreal number  from the a real number is similar to construction of a real from the rational numbers by means of equivalence class of Cauchy sequences (Staunton, 2013). To construct a hyperreals * , first we illustrate some notion of an ultrafilter, which will allow us to do a typical ultra power construction of the hyperreal numbers.
Suppose for = {1,2,…..} be the set of realvalued sequences, under point wise addition and multiplication. Let u = (ui), v = (vi) are elements in which defined as follows: However, uv and u⨀v are in , so its is closed under point wise addition and multiplication. So ( , , ⨀) is a commutative ring with identity sequence, however, satisfies all the properties of a field with identity, (1,11,1,…) and (0,0,0,…) = 0 and additive inverse consider, for example, the two sequences u = (0,1,0,1,…), v = (1,0,1,0…) neither of u or v equal to the zero. However, point wise multiplication would give us: Thus two nonzero elements u, v whose product is zero, are prevent the sequence to be an order field. To avoid this problem and to introduce equivalent relation which make into an order field and then ex tend it to the hyperreals * , we must construct a hyperreal number system * as an ultrapower of the real number system (Hurd and Loeb, 1985). To present an equivalence relation we need the notion of an ultrafilter to do so, first we must present a definition of a filter (Staunton, 2013).

Definition. 2.3.1. A Filter
Let  be a nonempty set of . A filter on  is a nonempty collection  of  having the following properties (Staunton, 2013): (i). The empty set  (ii). If U, V then UU (iii). If U and VU, then is cofinite or V

Definition. 2.3.2. A Free filter
If all elements of a filter are infinite sets then it said to be free (or non-principal).

Proposition. 2.3.1.
a. Every  filter contains the nonempty set 

Definition 2.3.3. An Ultra Filter
A filter  is said to be an ultrafilter denoted by  iff any subset U of  either U or U c  (not both by(i), (ii)) where: If  is an infinite set the collection yl = {AI: I-A is finite} is a filter called the cojnite or Frichet filter on .

Proposition. 2.3.2
Let U, V and U c  be a complement of U then the following properties are holds: iii. An ultraflter  is an ultrafilter on  iff  is amaximal proper filter

Definition. 2.3.4. The Fre'chet Filter
If  is an infinite set then collection: is called a Fre'chet filter or cognate.
The Fre'chet filter  is not an ultrafilter Moreover its proper iff  is infinite. An ultrafilter  on  is free if it contains . Hence a nonprincapl ultrafilter must contain all a finite sets. This is a critical property used in construction infinitesimals and infinitely large numbers. Here are some important properties of  (Staunton, (2013).

Definition.2.3.5. Free Ultra Filter
Combining definitions (2.3.3) and (2.3.4) we come up with the definition of a free ultrafilter.
Also  is said to be free if it contains the Fre'chet filter. A free ultrafilter  on  contain any finite set of . Moreover, all elements of the  are infinite sets (Davis, 2009).
The free ultrafilters doesn't always exist. Hence, an ultrafilters are important for the purpose of construction of hyperreal * .
An infinite sequences of real numbers are represent a free ultrafilter often use to give a rules for equality and identification, so we can come up with a mathematically consistent and sensible system of hyperreal numbers therefore in this way a hyperreal number can be generated. (ii) and (iii) for a filter. The equivalence relations are use the notation u  v.

An Equivalence Relation on Real Valued Sequence
The set can be divided into disjoint subsets (called equivalence classes) by the relation . Each equivalence class consists of all sequences equivalent to any given sequence in the class, therefore u and v are said to be in the same equivalence class iff u = v. Two sequences which differ at only a finite number of places are equivalent under  (Hurd and Loeb, 1985).

The Equivalence Classes on a Real Valued Sequence and an Ultrapower
In order to extend the real numbers system to the hyperreal * in an ultrapower concept we can use infinite sequences of real numbers (Davis, 2009) before doing so, we shall create a field of real-valued sequences, in which every standard real numbers are embedded as the corresponding constant sequence.
Let M denote the set of all the quivalence classes of in deuced by . The equivalence class containing a particular sequence u = (ui) is denoted by [u] We can define a relation  on by putting: When this relation holds it may be said that the sequences ui = vi possess same values at almost i.
Elements of M are called nonstandard or hyperreal number s and technically its known as an ultrapower (Goldblatt, 2012).

Lemma. 2.5.1
The relation  is an equivalence relation on a hyperreal . Let M denote the set of all equivalence classes of * in duced by . The equivalence class containing a particular sequence u = (ui) is denoted by [u] We now define an operations and relations which we will used it to make into an ordered field. . .

Remarks
is technically known as an ultra-power. We have used the same idea to define operations and relations which make into an ordered field (Goldblatt, 2012).
Logically for u, v we ca n replace the set, {r : ui = vi} by (u = v) thus u  v iff (u = v). Now since we have all the necessary tools so that we are ready to show that that * is order field.
Then (u⨀v = 1) is equal to w so: we want to show that * is a linearly ordered field with the ordering given by < .  Hence exactly one of: We want to show that only one of k, m and n is in , from the law of trichotomy in , we see that k  m  n = . Now one of l, m and n are in  also k c , m c , n c are in  also: which is a contradiction. Clearly the fact that * is totally ordered follows from the fact that is totally ordered and that  is an ultrafilter.
Thus, we have that * is a totally ordered field. Hence a hyperreal numbers is totally orderd field * numbers (Hurd and Loeb, 1985). The filed is embaded into * through a mapping that assign to each u , the hyperreal u *  * denoted by the equivalence class of the constant sequence with value u we shall identify u and u * . Now we want show that areal number system can be embedded isomorphic as linearly ordered sub field of * .
One can relate areal number u with constant sequence u = [u, u, u,……….] assign to the * elements. Define (u) = u * where, u * = [u] = [u, u, u,……….]. For u, v we have the following properties: The map : u  u * is an order preserving field isomorphism from to * .

Proposition. 2.5.3.
Given : Moreover  produces a proper extension G * G  with equality iff G is a finite set. Those properties imply that the image * is a nonstandard model in the formal logic sence which we will describe below.

Remark
Therefore we conclude the fact that:  The * , * , * and * are extensions of , , and * , respectively  The hyperreal extension *  preserves all order properties of an ordered fields, hence a real numbers form of a hyperreal numbers and the order relation. Therefore * is an ordered field extension of The following principle is a necessary for extending sequences and functions to the hyperreals (Davis, 2009).

The Extending Principle
Each function in the standard models can be extended it to a function acting on the corresponding nonstandard models. To be accurately for every real function f of one or more variables there is a corresponding hyperreal f * of the same numbers of variables, f * denoted the natural extension of f.
We can extend real-valued functions to hyperrealvalued functions in the following ways, let : If there is a function f: L where L , to extend the function f to the hyperreals, we have to define the extension of its domain L to a subset L of the hyperreal (Davis, 2009). We define the extension L * of a subset L of areal to be the set: Therefore L is the set of equivalence classes of sequences whose values range over the elements of L. Then the a function : f  can extend to a function * * * : f  therefore such extension processes same rules as the original functions and relations.

The Formal Language of Relational Structure
In this section we will give an elementary idea on the language theory of relational structure with a few examples.
We will use a formal logical symbol to express statements that were asserted to be true or false of the structure and * .

Definition. 3.1.1. (Relational Structure)
A relational structure is a system  = {, q, f} consists of a nonempty set , a collection of finitary relations q on  and a set f functions relations on .

Definition. 3.1.2
A language L is a set that including all logical symbols and quantifiers (including the equality sign and the parenthesis) and some arbitrary number of constants, variables, function symbols and relation symbols (Stroyan and Luxemburg, 1977).
In order to describe a formal language it is first necessary to describe the symbols of the language and then we can describe the process of forming sentences. each relational stricter  then L is language L is bases in the logical symbol. Therefore any statement that is expressible in logic structure is mentioning only standard numbers is true in if and only if it is true in * . The basic symbols is divided into two types: (1). The symbols consists of logical symbols which are common to any simple language and do not vary if the statements is changed as containing the following symbols: (2). The symbols in the second category depend on Q and will be called parameters. They consist of constant Symbols, Relation Symbols, function Symbols

The Formula and the θ-Term
There are expressions like composite functions in usual mathematical notation, constant, variable and function symbols is a string of symbols from the alphabet, they are special cases of terms we denoted L-term and they are defined inductively by the following laws:  Each constant symbol is an L-term.
i. Each variable symbol is an L-term.
ii. If f is the name of a function of n variables and 1,…,n are L-terms, then f(1,…,n) is an L-terms For example f (2, g(x, y)) and cos(x + y) are L-terms

Definition. 3.2.1. (A Closed Term)
A closed term is term which that made up of constant and functions symbols. It is undefined if it does not name anything.

Definition.3.2.2. (A formula)
If ,  are L-formula then it follows that:  If  is a formula so is ¬. If  and ψ are formulas, then so is ()  If  is a formula and x is a variable then (x) is also a formula  If  is L-formula and x is any variables symbol and P is subset of Q then, (xP), (xP) are L -formula Now we will give some basic concepts in mathematical logic.

Definition. 3.3.1. (A Sentences)
A sentence known as a formula in which all variables Are bounded. If the closed terms of the sentence are all defined then it has a fixed meaning and it is either true or false.

Definition. 3.3.2. (A Free Variables)
A free variable is a variable which obtained by Replacing any variable occurring in a statement by some constant to obtain another meaningful statement.

Definition. 3.3.4. (Bounded Variables)
A variable that is not free is called a bounded or dummy variable.

Definition. 3.3.5. (A Tomic Sentences)
An atomic sentences is a formula which has no variables and written are of the form Q(1,…,n), where Q is n-ary relation and (1,…,n ) resents terms' of the sentences. A sentences also may defined inductively (Goldblatt, 2012).

Definition 3.3.6. (A Simple Sentences)
A simple sentences known as a language takes two types denoted by, atomic and a compound sentence and consists of a basic and combinations symbols which defined as a string of symbols in a sentence.

An Atomic Relation
An atomic relations are simplest mathematical relations by which are meant relations containing neither logical connectives nor quantifiers, hence relations such, as (=, ,…etc.) (Staunton, 2013).
Atomic relations can be regarded as functions be {true, false}. A sentences are of the form Q1,…,n where Q the name of an n-ary relation and the  i (I = 1,…,n) are constant terms therefore 1,…,n are all closed i.e., there is no variables in the formula (Goldblatt, 1998).
An arbitrary statement which composed of a finite number of atomic relations and logical connectives can be define it inductively as follows, if  and  are two sentence, then the following are also sentences:

42
Not that all the logical connectors can be derived from logical symbols , ,  as follows:

Example. 3.4.1
The first order field axioms can be expressed as first order logic statements as follows:  Associativity Prosperity: We now will introduce the notion of the -transform of first-order L sentence which is useful tool for transforming L-sentences in areal to the * L  sentence in a hyperreal * .

The Truth Value of Sentences
Recall that a sentence is either true or false in the real number system. Let are two sentence with standard meaning of symbolic connectives , , , ,  we will present some rules that usually using for calculation a truth values of a sentences:   are true if  are true and  are true   are true if  are true ore  are true   are true if  are false     is true if and only if either  is false or else  is true     is true if and only if  and  either both true ore both false The mathematical formulation for statements associated with truth values rules able use to distinguish exactly which property is can transferable from to * and vice-versa.

The Transfer Principle
The transfer principle is the powerful tool that allows us to use the methods of non-standard analysis to prove results in standard analysis (Staunton, 2013).
The transfer principle states that" a formula is true on areal system if and only if the corresponding formula is true on * ".hence the transfer principle allow us to show that a hyperreal * has all the properties of and also we can prove theorems about by first proving them in * on the other words a transfer principle extends all a classical rules on a reals system to the hyperreal system which allow for easier and more intuitively natural proofs in a hyperreal system (Davis, 2009).

The -Transforms for First-Order Sentences Definition 4.1.1
The *-transform of a simple sentence Q in L-formula is the simple sentence Q * in * L  -formula obtained by starring all function and relation symbols in the sentence Q.
Thus, constructing a * -transform of a sentence L really just consists of putting a * on every term in Q, putting a * on any relation symbol in Q and putting a * on every set in Q acting as a bound on a variable (Davis, 2009;Keisler, 1976;Goldblatt, 2012).

Notation
Note that the * -transform arises by attaching the prefix to symbols but not attaching to variable symbols (Goldbring, 2014).
First we introduce a number of example which illustrate how we the a statements can be transforming.

Example. 4.1.1
Any positive real has areal n-th root for all n . This statements can be formulated as: Which is true. We can transform it to the true sentence: Which assert that, a hyperreal number has a hyperreal n-th root for all n * . This the a * -transform of a sentence (1) which is also true.

Example. 4.1.2
There does not any numbers x . such that x<1. This statement can formulated as follows: Which is true. We can transform it to the true sentence: Which assert that there are no number x of * smaller than 1.
From This example we conclude that a number of * / must be larger than all elements of , hence is infinitely large (unlimited).

Example 4.1.3
Archimedean property assert that, given any real numbers x there exists a natural number n (depending on x) such that x<n.
Archimedean property can expressing as follows: Which is true. The * -transform of (3) can gives by: Therefore since n is unlimited, then the statement ( 4) is true in * .
The * -transform of terms can be defined by induction on the formation of  as following laws:  If  is a constant or variable symbol then  * = .  If  = f(1,…,n) then =  * = f * (1 * ,…,n * ) A -transform of a sentences can be defined as follows:  Replacing each term  occurring in Q by Q *  Replacing the relation symbol g of any atomic formula occurring in Q by g *  Replacing the symbol g of any quantifier (xP) or (xP) occurring in Q by g *  The symbols <,>; will denote the corresponding relation and functions in in and * (Goldbring, 2014) Thus, constructing a * -transform of a sentence L really just consists of putting a * on every term in Q, putting a * on any relation symbol in Q and putting a * on every set in Q acting as a bound on a variable (Davis, 2009;Keisler, 1976;Goldblatt, 2012). For writing firstorder sentences, we can construct a method of transforming sentences in to sentences in * as follows, the *-transform  * of an L-term  which The following first-order of L-formula of the totally ordering field which mentioned in example (2.5.1) is equivalent to first order * L  as follows:  Associativity Prosperity: So the list of first-order sentences * L  above are true in * .
Recall that the -symbol can droped in the following case:  If the symbol refer to the transforms of well-known relations such as =, , <, , , ……etc.  If the symbol referring to well-known mathematical functions such as sin, cos, tan, cot, …….. etc.  If: XY, then f * : X * Y * and f * (x) = f (x) if xX.
Often the *-symbol in f * may be dropped  Consider addition in . Its * -transform is *addition in * and x+y = x + y if x, y . The symbol can safely be dropped Atomic relations are relations in which neither logical connectives nor quantifiers play a part, but only such relations as < or , etc. Consider first < in , leading to < * in * . Similarly as under e) we have that x < * y is equivalent to x < y if x, y and again the -symbol can safely be dropped.
The logical connectives (, , , , ) and both quantifiers (, ). For all of them the * -transform is identical to the inverse image, so that -symbol should be dropped.
The idea of constructing a * -transform such that the first-order of sentence would be true if and only if * L  is true, is called the transfer principle. Hence the * -transforms of first-order sentences, which we know to be true by the transfer principle (Goldblatt, 1998).
Thus the Transfer Principle asserts that every first order statement true over is similarly true over * and vice versa. This means that every statement is valid for areal if and only if the corresponding formula is valid on ahyperreal * , hence the transfer Principle asserts that every first order statement true over is similarly true over * and vice versa (Davis, 2009). This means that the truth of the statements follows by the transfer principle from the fact that the sentence is true in its standard structure. Now we will introduce some example.

Example. 4.2.1
The following first-order sentence which expressing the Archimedean property of the real numbers, using the mathematical logic it can be written as: By applying a transfer principle the equivalent firstorder of * L  -sentence given by:

Example.4.2.2
Consider the standard mathematical functions which given by:  (x ) cos(-x) = cos x  (x ) cosh x + sinh x = e x  (x, y ) log xy = log x + log y is true. Then it can be transferrable to a hyperreal * hence the following facts:  (x * ) cos(-x) = cos x  (x ) cosh x + sinh x = e x  (x, y ) log xy = log x + log y Also are true. Since areal number is an ordered field which we expressed it in finite number of L-sentence, by transfer principle we can conclude that the * -transform of these Lsentence are true hence these explain that * is an ordered field. So, instead of explicitly proving the ordered field axioms ordered field axioms, we can simply take the *transforms of list which we mentions in example (4.1.4) of first-order sentences of, that it is true by the transfer principle. We have thus proven that * is a totally ordered field without ever considering * as an ultrapower of , nor even doing a single ultrafilter calculation.

The Existential of Transfer Principle
The existential of transfer principle states that "If there exists a hyperreal number satisfying a certain property then there exists a real number with this property".
This principle can used to conclude that the original sequence must be bounded in areal .

Theorem. 4.3.1
If the extend hyperal sequence * * * :, u  is never takes infinitely large values then the extend sequence u * is bounded in .

Proof
Which prove that the sequence is bounded in .

Proposition. 4.3.3
If U and V are two sets in n then:   * =   (AB) * = U * V *  (UV) * = U * V *  (U') * = (U * )'  UV then U * V * which expresses the facts true in * (Goldblatt, 1998). Now we present theorem which also assert that a first-order formula in is true if and only if it is true in * , so its the transfer principle is direct consequence of.
That is Lo's' theorem which is also sometimes known as the fundamental theorem of Ultra products. We give its formal statement below.

Theorem 4.3.4 (Lo's' Theorem)
For any L formula Q(u1, u2…..um) and any r 1 , r 2 …..r m  n the sentence Q * ( The Lo´s' Theorem include transfer as special case because if Q is a sentence then it has no free variables so that Q(v1,...,vm) is just Q and likewise for Q * . Thus [Q(r 1 ,...,r m )] is in if Q is true and  otherwise, independently of sequences r j . Since  Lo's' Theorem in this case simply says, Q * is true if Q is true. Which is the transfer principle (Goldblatt, 1998;Staunton, 2013).