let s, s' be Complex_Sequence; :: thesis: ( s is convergent & s' is convergent implies s + s' is convergent )
assume A1: ( s is convergent & s' is convergent ) ; :: thesis: s + s' is convergent
then consider g being Element of COMPLEX 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) - g).| < p by Def4;
consider g' being Element of COMPLEX such that
A3: 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) - g').| < p by A1, Def4;
take g1 = g + g'; :: according to COMSEQ_2:def 4 :: thesis: 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 + s') . m) - g1).| < p
let p be Real; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.(((s + s') . m) - g1).| < p )
assume p > 0 ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.(((s + s') . m) - g1).| < p
then A4: p / 2 > 0 by XREAL_1:141;
then consider n1 being Element of NAT such that
A5: for m being Element of NAT st n1 <= m holds
|.((s . m) - g).| < p / 2 by A2;
consider n2 being Element of NAT such that
A6: for m being Element of NAT st n2 <= m holds
|.((s' . m) - g').| < p / 2 by A3, A4;
reconsider n = max n1,n2 as Element of NAT by FINSEQ_2:1;
take n ; :: thesis: for m being Element of NAT st n <= m holds
|.(((s + s') . m) - g1).| < p
let m be Element of NAT ; :: thesis: ( n <= m implies |.(((s + s') . m) - g1).| < p )
assume A7: n <= m ; :: thesis: |.(((s + s') . m) - g1).| < p
n1 <= n by XXREAL_0:25;
then n + n1 <= n + m by A7, XREAL_1:9;
then n1 <= m by XREAL_1:8;
then A8: |.((s . m) - g).| < p / 2 by A5;
n2 <= n by XXREAL_0:25;
then n + n2 <= n + m by A7, XREAL_1:9;
then n2 <= m by XREAL_1:8;
then |.((s' . m) - g').| < p / 2 by A6;
then A9: |.((s . m) - g).| + |.((s' . m) - g').| < (p / 2) + (p / 2) by A8, XREAL_1:10;
|.(((s + s') . m) - g1).| = |.(((s . m) + (s' . m)) - (g + g')).| by VALUED_1:1
.= |.(((s . m) - g) + ((s' . m) - g')).| ;
then (|.((s . m) - g).| + |.((s' . m) - g').|) + |.(((s + s') . m) - g1).| < p + (|.((s . m) - g).| + |.((s' . m) - g').|) by A9, COMPLEX1:142, XREAL_1:10;
hence |.(((s + s') . m) - g1).| < p by XREAL_1:8; :: thesis: verum