Конспект лекций по математическому анализу. Шерстнев А.Н. - 336 стр.

2,m . pOLOVIM X c = Tm Xnm0 (m). tOGDA fnXc =) fX c , TAK KAK jfn(x)Xc (x),
f (x)Xc (x)j < m1 (n  n0). zAMETIM DALEE, ^TO
                        [          \[
             (X m )c = ( Xnm )c =        fx : jfi(x) , f (x)j  m1 g:
                        n          n in

 pO\TOMU ESLI x 2 (X m)c , TO SU]ESTWU@T SKOLX UGODNO BOLX[IE n, PRI
KOTORYH jfn(x) , f (x)j  m1 , TO ESTX x 2 (X m)c ) fn (x) 6! f (x). w SILU
USLOWIQ TEOREMY (X m )c = 0. oTS@DA
        X = ([Tm Xnm0 (m)]c ) = (Sm(Xnm0 (m))c)  P (Xnm0 (m))c
             = P (X m nX m ) < P 2,m = : >
                                                     m
                  m          n0 (m)    m

   5. u P R A V N E N I E. eSLI fn 2 M (E; A) I fn ,!   P.W. f , TO fn ,!
                                                                       P.W. g
TTOGDA g  f .
   x206. sHODIMOSTX PO MERE
   1. pOSLEDOWATELXNOSTX fn IZMERIMYH FUNKCIJ (NA PROSTRANSTWE S
-KONE^NOJ MEROJ) NAZYWAETSQ
                   
                                SHODQ]EJSQ PO MERE K IZMERIMOJ FUNKCII
f (OBOZNA^ENIE fn ,!   f ), ESLI DLQ WSQKOGO " > 0:
                     n fx : jfn (x) , f (x)j  "g = 0:
                   lim
   2. pUSTX fn | POSLEDOWATELXNOSTX IZMERIMYH FUNKCIJ NA PROSTRAN-
STWE E S KONE^NOJ MEROJ , PRI^EM fn ,!  P.W. f . tOGDA fn ,!
                                                             
                                                               f.
  w SILU 205.3 f IZMERIMA. pROWERIM, ^TO Xn (") ! 0 (n ! 1), GDE
()                 Xn (")  fx : jfn(x) , f (x)j  "g:
pUSTX X = T S X ("). eSLI x 2 X , TO x PRINADLEVIT BESKONE^NO
              1
                       k
             n=1 kn
MNOGIM Xn (") I, SLEDOWATELXNO, fn(x) 6! f (x). pO\TOMU
          X  fx j fn (x) 6! f (x)g; fx j fn (x) 6! f (x)g = 0:

                                      336