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1 Classification of injective mappings and number sequences ALEXANDER M. SUKHOTIN, TATIJANA A. SUKHOTINA Department of Higher Mathematics Tomsk Polytechnic University 30, Aven. Lenin, Tomsk, 634050 Russia Abstract: New concepts are entered in the theory of injective mappings and in the theory of numerical sequences such as: a precise pair of variables, a divergent sequence, a convergent in itself sequence etc. The new methodological approach has allowed to classify injective mappings and numerical sequences and to prove some paradoxical from the classical point of view the statements on the analysis: the existence of infinity large Cauchy’s sequences has made possible and necessary the introduction of infinity large numbers. Key-Words: Exact surjectivity, Antisurjectivity, Number Sequence, Divergent Sequence, Infinitely Large Numbers. 1 Introduction The Great scientist of XVII century G. Galilei, having discovered that the quantities of natural numbers and their quadrates are equal, has bequeathed to the successors to be very cautious at an operation with infinite amounts: "… the properties of equality, and also greater and smaller values have no the places there, where the matter goes about the infinity, and they can be applied only to finite amounts" [1, p. 140-146]. The ignoring of this warning has entered into the mathematical folklore some false hypotheses together with its proofs that contain incorrect reasoning. These and contiguous by them problems were as a subject of learning in this work. The new procedure enabled us to overcome the above difficulties. The injective mappings : N N and the properties of numerical sequences have been analyzed in this work. Classification of the investigated objects has become one of the results of our research. This theory is borne out by the facts too. 2 Problems Formulation For finite sets A and B the check of mapping : A B surjectivity does not cause difficulties. On the contrary, the similar procedure for mappings of infinite sets is not such trivial. Injective mappings : N N , except for obvious antisurjective such, as ψ(n) n 1 , are considered bijective by default in the traditional mathematical texts. The proof of surjectivity criteria for injective mappings : N N and their classification made up the first problem, which was solved in this paper. The second problem – the research of properties of numerical sequences and their classification has been solved due to the introduction of positive definition of a divergent numerical sequence. The main result of these researches has been formulated in the following form: Theorem 1. Any fundamental number sequence (а) satisfies to the following condition: ( 0 n() N ) : ( n n() an1 an ), (1) or, that is the same, lim (a n 1 a n ) 0 . (2). 3 Solutions of problems 3.1 About properties of injective mappings NN The infinity of set N : (1, 2, , n, ) of natural numbers is understood in connection with a principle of a mathematical induction as unbounded possibility of transition from (n) to (n+1). More common phrases "at a passage to the limit in F(n)" and "at n in F(n)" mean the following: n N F n F n 1 . (3) 2 The principle of the passage to the limit (3) and an uniform ordering of set of natural variables make possible the introduction of the following concept. Definition1. The pair (n, m) of variables, n, mN, is named as a precise pair at n , if (4) ( p, n( p) N ) n n( p) m n p . Let there is an infinite sequence of natural numbers =( 1, n1 , n2 ,, ni , ) and let ni ni 1 . The sequence (the splitting ()) divides set N into the segments i (ni1 1,, ni ), n0 0 , N i , i I N . The injective mapping : N N and splitting () induce three sequences: i , d i and d i of the non-negative integers under the following formulas: i ki ni , ki max{(n) : n ni } , i I N , (5) d i Di k : n ni k (n) ni , i I N (6) d i Di k : k ni , n ni k (n) , (7). i I N . From the condition (5) follows that ki ki 1 . Figure 1 illustrates the mapping : N N , where Statement 2. If such splitting () of sets N on pieces i and number C exist for an injection : N N , that for i I N i C , then for any number С>0 new splitting ( ) of set N can be received by means of corresponding enlargement of pieces i , that inequalities i C C and С<inf{ i , i I N } will be fair for pieces of splitting ( ). Classification of injective mappings : N N is carried out to following two properties: 1) a triviality, not trivial boundedness, unboundedness of the sequence ( i ) , 2) fullfilment or default of a condition i N j : Di Di j Conditions (5)(7) allow to separate all injective mappings : N N on three not crossed classes: Definition 2. Injective mapping : N N is determined as exact ones, if there is a sequence =( 1, n1 , ) such, that i N i =0. Definition 3. Injective mapping : N N is determined as potentially surjective ones, if there are a sequence =( 1, n1 , ) and number C >0 such, that almost i N 0< i C and besides i N j : Di Di j . (m) p . Definition 4 Injective mapping : N N is determined as antisurjective ones, if at least one of two following conditions is carried out: 1) С>0, =(1, n1 , ) i(, C ) : i (, C ) >С, 2) =( 1, n1 , ) {С(), i () , n }: n, (9) n> n N: N k : k (n), k ni ( ) =С(). Fig. 1 Here , s 2 , s3 Di , i p2 ni . From determining conditions (5)(7) follows, that 0 d i i , d i = d i . It is easy to prove the following Statement 1. Sequences ( i ) and (d i ) , determined in any pair (, ), satisfy for almost i I N to one and only to one of three following conditions: 1) ( i 0) (d i 0) , p2 Di 2) (d i ) ( i ) , (8) 3) C1 N: (0 i C1 ) (d i C2 C1 ) . The following below statement is a consequence of conditions (8): The necessary criterion exact or potential surjectivity of injections : N N is formulated on the basis of the classification given above as follows: Theorem 2. Injective mapping : N N is exact or potentially surjective ones in only case when the following below two conditions have been satisfied: 1) =( 1, n1 , ), ni N , C >0: i N i C (10) 2) i N j : Di Di j . Theorem 3. For injective mappings : N N of first two classes the following below limiting equality are fair: ~ : 1 , ~( ) ( ) a) lim i i i i i (11) b) lim n : n k 1 , if this limit exists. (12) 3 ● a) There is always ( i ) i for exact surjective injective mappings : N N and, hence, the conditions (11) are fair for such mappings. Generally, the opportunity of construction of splitting ( ) of set N for the given injection : N N , which provides the existence of corresponding limit (11), follows out of Statement 2. b) We shall assume opposite lim n : n k , 1<k<. Then there is a number n such, that n (1 )n >n for almost all n> n and the any , k 1 0 . Hence, n n n , that means the unboundedness of sequence ( i ) , that contradicts to the condition i < C of the Theorem 2. The similar conclusion can be received for function 1 : N N , if the k satisfies to the inequality 0<k<1 of (12).■ The statement following below is consequence of (10) for infinite subsets of set N (comp. [2, p. 20]): Theorem 4. A bijection does not exist between set N of natural numbers and its own subset АN. Theorem 4 can be proved with the help of the following below Theorem 5 (see [3]). Theorem 5. Let A and B be own subsets of set N also there is an injection : A B . Then this injection : A B can be continued up to bijection : N N . The opportunity of such continuation is proved by means of consecutive designing of points of set N \ ( A) on a diagonal of set N N and on the diagram of function : A B (see also Example 4 lower). Theorem 6. The equivalence ( rn 0)( a n 0) is fair for any number series (a) m k m of a number series (a). If Rk ,m S m S k , k m then the remainder rk of series (a) is determined by equality rk lim Rk ,m . Therefore, m lim rk lim ( lim Rk ,m ) . k m (13) Let formula m ζ (k ) determine an injection ζ : I N , I N . Then it is possible to consider by virtue the first of conditions (10) of Theorem 2 that m k δ i C in the limit condition (13). ai =0, because a i 0. 3.2 Examples Example 1 (G..Galileja's paradox [1]). It is obvious, that a mapping : N N defined by the formula (n) n 2 , satisfies to both conditions (9). Such mappings can be named as total antisurjective ones. Example 2 (to Th. 3). Let for some splitting () of set N on pieces i , i N , a mapping : N N be the change of the order on the pieces i , i N , on opposite ones. Then this mapping belongs to the first class since i I N δ i 0 . If splitting () is limited ones, i.e. C : i I i C , then lim ((n) / n) 1 (see (12)). If splitting () is not limited, then does not exist, and lim( (n) / n) ~ lim ( ( ) / ) 1 (see (11)). i i Example 3 (to Th. 6) It is know harmonic series ai is defined by formula: a n n 1 . For this 1 m series (see [4, p. 319]) S m = n 1 =lnm+ C e + m , 1 k S k = n 1 =lnk+ C e + k , where Ce is Euler’s 1 constant and n 0. Therefore, Rk ,m ln( m k ) γ m γ k . lim rk = lim (ln( (k p) k ) γ k p γ k ) =0 for harmonic series. k k p k i k 1 1 k k Hence, both S m a n and S k a n be the partial sums 1 lim rk lim ( lim S m S k ) lim ai . а) The implication (( rn 0)( a n 0) follows from rn rn 1 a n 1 . b) Further, let a n 0, let Therefore, it is possible to consider a pair (k , m) in (13) as a precise pair (see (4)), therefore, Example k 4. k (to Th. 5). Let A N \ {1} , B N \ {1, 2} and let φ : A B be an injection which is defined by the formula: (n) n 1 . If Ap (2, 3, ..., p) A and B p (3, 4, ..., p 1) B then ( Ap ) B p . Further, let C1 A=(2, 3, 4, …, m, …), C 2 (3, 2, 4, …, m, …), …, C k (3, 4, 5, …, k+1, 2, k+2, k+3, …), k A , be the simply ordered subsets of the set N and let the bijection n : Cn Cn1 be the transform of a pair (2, n+2) into ones (n+2, 2) and q C n \ (2, n 2) n (q) q C n1 , n N . 4 Now we shall determine a bijection n : C1 C n recurrently by means the rules: 1 1 , n N , n1 n1 n , i.e. p N ψ n ( An 2 p ) (3, 4, ..., n 2, 2, n 3, ..., n 2 p) , ψ n ( An 2 p ) B. By virtue of (3), we determine a bijection ψ : A A by means the formula: ψ lim ψ n . At last, we determine a bijection : N N by 1, if n 1, following way: ψ(n) ψ (n), if n 1. It is easy to see what p A ( Ap ) B p , i.e. the restriction of the bijection : N N on each subset Ap is a bijection between sets Ap and B p . Therefore, the bijection : N N is a continuation of the injection : A B . Besides, ψ( A) A φ( A) =B, because 2B. 3.3 Properties and classification of number sequences A sequence (а), (а) ( an ) ( a1 , a2 , , an , ), where nN, a n R , is named a convergent to number a R sequence and there are speaking in this case that this sequence converges to number a, if a following below condition (14) is fulfilled [5, S. 46], [6, S. 63]: 0 n N : (n n ) an a . (14) The number sequence (а) is named a fundamental sequence (regular sequence, Cauchy's sequence (CS)) (see [7, p. 355], comp. with [6, S. 81, 8586]), if lim (am an ) 0 . (15) min( m ,n ) It is easy to prove that the limiting equality (15) is equivalent “on the language (, n())” to following condition: ( 0 n() N ) : ( (n, m) min{ n, m} n(ε) a m a n ε). (16) It is possible to consider a pair (m, n) of variables n and m in (15) as a precise ones (see (4)): The ( p, n( p) N ) n n( p) m n p . validity of Theorem 1 follows from both the (15) and the (4) at p=1. Definition 5. A number sequence (а) is named "convergent in itself" ones (CIS), if (2) is fair or, that is same, if (ε 0 n(ε) N ) : ( n n() a n1 a n ). (17) The term "CIS" has been used in [5, S. 51] as a synonym for "CS". It is obvious, that the convergent to number a sequence (а) will be and CIS too. The inverse statement has no a place generally. Theorem 7. The following below limit equality is fulfilled for a convergent (or convergent in itself) number sequence at any natural number p (18) lim (an p an ) 0 . The validity of Theorem 7 follows from conditions (14), (or from (17), accordingly) and from the following equality: (a n p a n ) (a n p a n p 1 ) (a n p 1 (19) a n p 2 ) ... (a n 2 a n 1 ) (a n 1 a n ). Definition 6. The number sequence (a) is termed a divergent ones (DS), if there are two infinite subsequences ξ 1 , ξ 2 N , ξ 1 ξ 2 such, that the following condition ( 0 , n N ) : (m, k ) (1 , 2 ) m, k n (20) am ak is fulfilled (comp. with [5, p. 46]). The following statement is fair by virtue of conditions (15), (16) and (20): Theorem 8. Any numerical sequence is either regular sequence, or divergent sequence, i. е. (a) (a){CS}{DS}, {CS}{DS}=, (21) The direct proof of Theorem 1 is stated below. We suppose the opposite judgment that there is a sequence convergent in itself but ones is not a sequence will be divergent, that means the condition (20) is fair: ( 0 , n N : (m, k ) (1 , 2 ), m, k n : am ak (20) By virtue of infinity of sequences 1 , 2 there are as is wished many pairs (m , k ) (1 , 2 ) , m , k n among all pairs (m, k)( 1 , 2 ) such, that for some q0 N m k q0 and, for example, m k q0 . Then by virtue (20) a m a k a k q a k . (22) 0 Let q0 1 at (22) then by virtue of conditions (17) - (19), which are fair for any СIS, we have ( 0 n() N ) : ( n n() an1 an ), (23) then an inconsistency of inequalities (22) and (23) proves the theorem. Let 1 q 0 p , pN in (21). In this case by virtue of equality (18), which is fair for any CIS, we have ( 0 n (, p) N ) : 5 ( n~ n (, p) an~ q0 an~ ) . (24) ~ k ) is not Let n sup{n , n (, p)} . Any pair ( n, exact ones (see (4)) but by virtue of that the n~ accepts all values n N , n> n , there is, at least, ~ k ), n, ~ k > n one pair (n~ , k ) from all pairs ( n, 0 0 such, that n~0 k0 . Then the required contradiction follows out from the inequalities (20) and (24): ak q ak 0 0 k0 n~0 0 an~0 q0 an~0 . A sequence (a) is termed as limited ones, if С>0 n N a n <C. If this condition is not carried out, the number sequence (а) is termed as unlimited. Example 5. Let the sequence (s) determine for all n N by the formula: s n sin n , then it is limited and divergent ones. We shall prove below that the sequence (s) hasn’t any convergent subsequence. We shall assume opposite that ( N , a R , a 1 ) and lim a k a . Let b arcsin a . k By virtue of lim a k a we have k lim (sin k sin b) 0 , k k b k b or lim 2 sin cos 0 . This limiting k 2 2 equality will take place, if at least one of two limiting equalities k b k b lim sin 0 , lim cos 0 2 k k 2 is fair. The last conditions are equivalent to corresponding limiting equalities at relevant integer numbers z k and wk : 1 2zk lim k , k 0 , k k 1 2 wk lim k , 0. (25) k k k Every one from the equalities (25) contradicts to transcendence of number π [8, p. 5]. The example 5 let to speak, that the following below judgment is fair: Statement 3. Not any limited sequence contains a convergent subsequence. Definition 7. The number sequence (a) is termed total divergent ones, if a condition ( 0 , n () N ) : (m, k ) (1 , 2 ), m, k n , 1 am ak is carried out for any two infinite subsequences 1 , 2 N , 1 2 . Example 6. The trivial total divergent sequence is sequence determined for all n N by the formula: a n (1) n n . 3.4. Infinitely large numbers We select a class of «infinitely large sequences at the limit» from a set of unlimited sequences by means of following below definition. Definition8. We name a limiting value of each unlimited CS (a) as infinitely large number (ILN) determined by this CS (a) and the unlimited CS is termed as infinitely large CS (ILCS). And we shall (a) too. write a We shall designate the set of all ILN by symbol and we shall attribute the set R Ω as a symbol ~ . Further, we shall consider by definition that R (a) (a) . A set (, ) will be termed as a set of infinitely large elements. Now we shall enter one more symbol Δ ~ Rˆ R {, } . The class of unlimited sequences divides concerning convergence-divergence on six subclasses (compare with [6, S. 64, 65]): 1) The sequences convergent to appropriate concrete infinitely large number a (a) , 2) The sequences divergent to plus-infinity, 3) The sequences divergent to minus-infinity, 4) The divergent sequences having at least one subsequence, convergent to finite number, 5) The divergent sequences having at least one subsequence, convergent to some ILN, 6) The total divergent sequences. The divergent sequences of the second and third kind can be classified by the method similar ones, specified in [9, p. 26-28]. Below we consider some properties of ILCS. Statement 4. The sequence ( an ), determined n N by the formula an (n)1 0 , is the CS. n 1 a n1 a n (n 1)1 n1 (n 1) n n 1 n 1 0. n n n n Statement 4 supposes the following generalization. Theorem 9. Unlimited differentiated in ~ converges to corresponding function f : Rˆ R ILN ( f ) if and only if f () 0 . Transition to limit in Lagrange’s formula for function f written below: f (n 1) f (n) f ()((n 1) n), n n 1 , takes up the proof of Theorem 9. Theorem 9 helps to prove the following below Statement 5. The sequence ( an ), determined for 6 all n N by the formula: an Cn1 (ln n) , at 0, , C R , is ILCS. Statement 5 allows to formulate following below Statement 6. There is not neither greatest, nor least number in the set of infinitely large ones. Example 7. Let for n N n en == p 1 = ln n Ce n , where n 0 and 1 C e is Euler’s constant [4, p. 270]. By virtue both Theorem 6 and Example 3 a harmonious series ~ n 1 converges in R to ILN designated here as a symbol e . The following properties for limiting values of unlimited functions f (), g () Ω ~ f , g : Rˆ R turn out from Theorem 9. Statement 7. Cf () Ω , С≠0, Cf () g () Ω , C R . Statement 8. f () f () g () 0, 1 (( f () g ()) R). g () .Statement 9. f ' o ( g ) 1 , g ' o(( f ) 1 ) f () g () Ω . Statement 10. f O( g 2 f ( ) ) Ω . g ( ) Statement 11. If C R and C 0 then 1 C. f () The ILN have been termed in the non-standard analysis (see [2, p. 21 and further]) as non-standard, impracticable, actually infinite large or inaccessible numbers. References: 1] Galilei G. Selected Work, V. 1, Moscow: Science, 1964. (in Russian) [2] Gordon E.I., Kusraev A. G., Kutateladze S. S., Infinitesimal calculus, V. 1, Novosibirsk: Institute of Mathematics, 2001. (in Russian) [3] Sukhotin A.M., About alone false promise, Ukrainian mathematical congress-2001: International conference on functional analysis, August 22-26, 2001, Kyiv, Ukraine: Abstract.-Kyiv: IM NAS Ukraine, 2001.-P. 92. [4] Fikhtengolz G. M. Course of differential and integral calculus, V. 2, Moscow: Science, 1966.(in Russian) [5] Rudin W., Analysis, Weinheim: Pnisik-Verlag, 1980. [6] Knopp K., Theorie und Anwendung der unendlichen Reihen, Berlin: Springer-Verlag, 1922. [7] Weisstein Eric W., CRC Concise Encyclopedia of Mathematics, London-New York: ChapmanHall/CRC, 2002. [8] Baker A., Transcendental number theory, Cambridge: Cambridge University Press, 1999. [9] Sukhotin A.M., Alternative analysis principles, Tomsk: TPU Press, 2002