let s be Complex_Sequence; :: thesis: ( s = s1 + s2 implies s is convergent )
assume A1: s = s1 + s2 ; :: thesis: s is convergent
consider g being Complex such that
A2: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((s1 . m) - g).| < p by Def5;
consider g9 being Complex such that
A3: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((s2 . m) - g9).| < p by Def5;
take g1 = g + g9; :: according to COMSEQ_2:def 5 :: thesis: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((s . m) - g1).| < p
let p be Real; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
|.((s . m) - g1).| < p )
assume A4: p > 0 ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((s . m) - g1).| < p
then consider n1 being Nat such that
A5: for m being Nat st n1 <= m holds
|.((s1 . m) - g).| < p / 2 by A2;
consider n2 being Nat such that
A6: for m being Nat st n2 <= m holds
|.((s2 . m) - g9).| < p / 2 by A3, A4;
reconsider n = max (n1,n2) as Element of NAT by ORDINAL1:def 12;
take n ; :: thesis: for m being Nat st n <= m holds
|.((s . m) - g1).| < p
let m be Nat; :: thesis: ( n <= m implies |.((s . m) - g1).| < p )
assume A7: n <= m ; :: thesis: |.((s . m) - g1).| < p
n2 <= n by XXREAL_0:25;
then n + n2 <= n + m by A7, XREAL_1:7;
then n2 <= m by XREAL_1:6;
then A8: |.((s2 . m) - g9).| < p / 2 by A6;
m in NAT by ORDINAL1:def 12;
then A9: |.(((s1 + s2) . m) - g1).| = |.(((s1 . m) + (s2 . m)) - (g + g9)).| by VALUED_1:1
.= |.(((s1 . m) - g) + ((s2 . m) - g9)).| ;
n1 <= n by XXREAL_0:25;
then n + n1 <= n + m by A7, XREAL_1:7;
then n1 <= m by XREAL_1:6;
then |.((s1 . m) - g).| < p / 2 by A5;
then |.((s1 . m) - g).| + |.((s2 . m) - g9).| < (p / 2) + (p / 2) by A8, XREAL_1:8;
then (|.((s1 . m) - g).| + |.((s2 . m) - g9).|) + |.(((s1 + s2) . m) - g1).| < p + (|.((s1 . m) - g).| + |.((s2 . m) - g9).|) by A9, COMPLEX1:56, XREAL_1:8;
hence |.((s . m) - g1).| < p by A1, XREAL_1:6; :: thesis: verum