Download Classification of injective mappings and numerical sequences

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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() an1  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
NN
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.
Definition1. The pair (n, m) of variables, n, mN,
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  (ni1  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 АN.
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  Cn1 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 n1 , n  N .
4
Now we shall determine a bijection  n : C1  C n
recurrently by means the rules: 1  1 ,
n N ,
 n1  n1   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 2B.
3.3 Properties and classification of number
sequences
A sequence (а), (а)  ( an )  ( a1 , a2 , , an ,  ),
where nN, 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 n1  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() an1  an  ),
(23)
then an inconsistency of inequalities (22) and (23)
proves the theorem.
Let 1  q 0  p , pN 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.
Definition8. 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 n1  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:
ChapmanHall/CRC, 2002.
[8] Baker A., Transcendental number theory,
Cambridge: Cambridge University Press,
1999.
[9] Sukhotin A.M., Alternative analysis principles,
Tomsk: TPU Press, 2002