let X be Banach_Algebra; for z being Element of X
for s being sequence of X st s is convergent holds
lim (z * s) = z * (lim s)
let z be Element of X; for s being sequence of X st s is convergent holds
lim (z * s) = z * (lim s)
let s be sequence of X; ( s is convergent implies lim (z * s) = z * (lim s) )
assume A1:
s is convergent
; lim (z * s) = z * (lim s)
set g1 = lim s;
set g = z * (lim s);
A2:
0 + 0 < ||.z.|| + 1
by NORMSP_1:4, XREAL_1:8;
A3:
0 <= ||.z.||
by NORMSP_1:4;
A4:
now for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
||.(((z * s) . m) - (z * (lim s))).|| < plet p be
Real;
( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
||.(((z * s) . m) - (z * (lim s))).|| < p )assume
0 < p
;
ex n being Nat st
for m being Nat st n <= m holds
||.(((z * s) . m) - (z * (lim s))).|| < pthen A5:
0 < p / (||.z.|| + 1)
by A2, XREAL_1:139;
then consider n being
Nat such that A6:
for
m being
Nat st
n <= m holds
||.((s . m) - (lim s)).|| < p / (||.z.|| + 1)
by A1, NORMSP_1:def 7;
take n =
n;
for m being Nat st n <= m holds
||.(((z * s) . m) - (z * (lim s))).|| < plet m be
Nat;
( n <= m implies ||.(((z * s) . m) - (z * (lim s))).|| < p )assume
n <= m
;
||.(((z * s) . m) - (z * (lim s))).|| < pthen A7:
||.((s . m) - (lim s)).|| < p / (||.z.|| + 1)
by A6;
0 <= ||.((s . m) - (lim s)).||
by NORMSP_1:4;
then A8:
||.z.|| * ||.((s . m) - (lim s)).|| <= ||.z.|| * (p / (||.z.|| + 1))
by A3, A7, XREAL_1:66;
||.(z * ((s . m) - (lim s))).|| <= ||.z.|| * ||.((s . m) - (lim s)).||
by LOPBAN_3:38;
then A9:
||.(z * ((s . m) - (lim s))).|| <= ||.z.|| * (p / (||.z.|| + 1))
by A8, XXREAL_0:2;
A10:
||.(((z * s) . m) - (z * (lim s))).|| =
||.((z * (s . m)) - (z * (lim s))).||
by LOPBAN_3:def 5
.=
||.(z * ((s . m) - (lim s))).||
by LOPBAN_3:38
;
0 + ||.z.|| < ||.z.|| + 1
by XREAL_1:8;
then A11:
||.z.|| * (p / (||.z.|| + 1)) < (||.z.|| + 1) * (p / (||.z.|| + 1))
by A3, A5, XREAL_1:97;
(||.z.|| + 1) * (p / (||.z.|| + 1)) = p
by A2, XCMPLX_1:87;
hence
||.(((z * s) . m) - (z * (lim s))).|| < p
by A10, A9, A11, XXREAL_0:2;
verum end;
z * s is convergent
by A1, Th4;
hence
lim (z * s) = z * (lim s)
by A4, NORMSP_1:def 7; verum