let X be Complex_Banach_Algebra; :: thesis: for z being Element of X
for s being sequence of X st s is convergent holds
z * s is convergent
let z be Element of X; :: thesis: for s being sequence of X st s is convergent holds
z * s is convergent
let s be sequence of X; :: thesis: ( s is convergent implies z * s is convergent )
A1: 0 <= ||.z.|| by CLVECT_1:105;
assume s is convergent ; :: thesis: z * s is convergent
then consider g1 being Point of X such that
A2: for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
||.((s . m) - g1).|| < p by CLVECT_1:def 15;
take g = z * g1; :: according to CLVECT_1:def 15 :: thesis: for b₁ being Element of REAL holds
( b₁ <= 0 or ex b₂ being Element of NAT st
for b₃ being Element of NAT holds
( not b₂ <= b₃ or not b₁ <= ||.(((z * s) . b₃) - g).|| ) )
let p be Real; :: thesis: ( p <= 0 or ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= ||.(((z * s) . b₂) - g).|| ) )
assume A3: 0 < p ; :: thesis: ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= ||.(((z * s) . b₂) - g).|| )
A4: 0 + 0 < ||.z.|| + 1 by CLVECT_1:105, XREAL_1:8;
then consider n being Element of NAT such that
A5: for m being Element of NAT st n <= m holds
||.((s . m) - g1).|| < p / (||.z.|| + 1) by A2, A3;
take n ; :: thesis: for b₁ being Element of NAT holds
( not n <= b₁ or not p <= ||.(((z * s) . b₁) - g).|| )
let m be Element of NAT ; :: thesis: ( not n <= m or not p <= ||.(((z * s) . m) - g).|| )
assume n <= m ; :: thesis: not p <= ||.(((z * s) . m) - g).||
then A6: ||.((s . m) - g1).|| < p / (||.z.|| + 1) by A5;
A7: ||.(z * ((s . m) - g1)).|| <= ||.z.|| * ||.((s . m) - g1).|| by CLOPBAN3:38;
0 <= ||.((s . m) - g1).|| by CLVECT_1:105;
then ||.z.|| * ||.((s . m) - g1).|| <= ||.z.|| * (p / (||.z.|| + 1)) by A1, A6, XREAL_1:66;
then A8: ||.(z * ((s . m) - g1)).|| <= ||.z.|| * (p / (||.z.|| + 1)) by A7, XXREAL_0:2;
A9: ||.(((z * s) . m) - g).|| = ||.((z * (s . m)) - (z * g1)).|| by CLOPBAN3:def 4
.= ||.(z * ((s . m) - g1)).|| by CLOPBAN3:38 ;
0 + ||.z.|| < ||.z.|| + 1 by XREAL_1:8;
then A10: ||.z.|| * (p / (||.z.|| + 1)) < (||.z.|| + 1) * (p / (||.z.|| + 1)) by A1, A3, XREAL_1:97;
(||.z.|| + 1) * (p / (||.z.|| + 1)) = p by A4, XCMPLX_1:87;
hence not p <= ||.(((z * s) . m) - g).|| by A9, A8, A10, XXREAL_0:2; :: thesis: verum