let X be Banach_Algebra; :: thesis: 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; :: thesis: for s being sequence of X st s is convergent holds
lim (z * s) = z * (lim s)
let s be sequence of X; :: thesis: ( s is convergent implies lim (z * s) = z * (lim s) )
assume A1: s is convergent ; :: thesis: 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 :: thesis: 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))).|| < p
let p be Real; :: thesis: ( 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 ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
||.(((z * s) . m) - (z * (lim s))).|| < p
then 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; :: thesis: for m being Nat st n <= m holds
||.(((z * s) . m) - (z * (lim s))).|| < p
let m be Nat; :: thesis: ( n <= m implies ||.(((z * s) . m) - (z * (lim s))).|| < p )
assume n <= m ; :: thesis: ||.(((z * s) . m) - (z * (lim s))).|| < p
then 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; :: thesis: verum
end;
z * s is convergent by A1, Th4;
hence lim (z * s) = z * (lim s) by A4, NORMSP_1:def 7; :: thesis: verum