Конспект лекций по математическому анализу. Шерстнев А.Н. - 463 стр.

nETRUDNO WIDETX, ^TO f 2 . dLQ ZAWER[ENIQ PROWERKI AKSIOMY (V) NUVNO
LI[X UBEDITXSQ, ^TO (f ) | ISKOMAQ NAIMENX[AQ MAVORANTA MNOVESTWA E .
iTAK, NUVNO USTANOWITX, ^TO
   (A) (f ) 2 F ,
   (B) ESLI (g ) | MAVORANTA E , TO (f )  (g ).
   pROWERIM (A). pUSTX, NAPROTIW, (f ) | NE MAVORANTA E . tOGDA 9(g) 2
E ((f ) < (g)). sLEDOWATELXNO, SU]ESTWU@T " > 0 (" 2 Q) I N1 2 N TAKIE, ^TO
" < g (n) , m0:m1 m2 : : :mn,1 (n > N1). pUSTX N2 2 N TAKOE, ^TO 10,N2 < ", I
PUSTX N = maxfN1; N2g. tOGDA DLQ n > N + 1 (SM. (9))
          g(n) , qN = g(n) , m0 :m1m2 : : :mN , 10,N
                     = g (n) , m0 :m1m2 : : :mn,1 + m0:m1 m2 : : :mn,1
                     , m0:m1m2 : : :mN , 10,N
                     > " + 0:0 : : :mN +1 : : :mn,1 , 10,N  " , 10,N2 ;
TO ESTX (qN ) < (g) I ZNA^IT (qN ) | NE MAVORANTA E . |TO PROTIWORE^IT
(9).
    pROWERIM (B). pUSTX, NAPROTIW, (g) | MAVORANTA E , PRI^EM (g) < (f ).
tOGDA SU]ESTWU@T " > 0 (" 2 Q) I N1 2 N TAKIE, ^TO " < m0:m1m2 : : :mn,1 ,
g (n) PRI n > N1 . wYBEREM N2 2 N TAKOE, ^TO 10,N2 < ", I PUSTX N =
maxfN1; N2g. tOGDA DLQ n > N + 1
       m0 :m1m2 : : :mN , g(n) = m0:m1m2 : : :mN , m0 :m1m2 : : :mn,1
                                 + m0:m1m2 : : :mn,1 , g (n)
                                 > ,0:0 : : :mN +1 : : :mn,1 + "
                                 > " , 10,N  " , 10,N2 :
oTS@DA -RACIONALXNOE ^ISLO (m0:m1m2 : : :mN ) | MAVORANTA E , ^TO PRO-
TIWORE^IT (9). iTAK, MODELX R POSTROENA.
    w ZAKL@^ENIE OTMETIM E]E DWA WAVNYH FAKTA.
    24. t E O R E M A [EDINSTWENNOSTI]. pUSTX (R; +; 
; ) I (R  b ; +b ; b
; b ) |
DWE MODELI MNOVESTWA DEJSTWITELXNYH ^ISEL. tOGDA SU]ESTWUET BIEKCIQ
j : R ! Rb TAKAQ, ^TO DLQ L@BYH ; 2 R
    (A) j ( + ) = j ( )+b j ( ),
    (B) j ( 
 ) = j ( )b
j ( ),
    (W)  TTOGDA j ( )b j ( ).


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