let s, s9 be Complex_Sequence; :: thesis: ( s is convergent & s9 is convergent implies lim (s + s9) = (lim s) + (lim s9) )
assume that
A1: s is convergent and
A2: s9 is convergent ; :: thesis: lim (s + s9) = (lim s) + (lim s9)
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 + s9) . m) - ((lim s) + (lim s9))).| < p

proof

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 + s9) . m) - ((lim s) + (lim s9))).| < p )
assume 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.(((s + s9) . m) - ((lim s) + (lim s9))).| < p
then A4: 0 < p / 2 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) - (lim s)).| < p / 2 by A1, Def5;
consider n2 being Element of NAT such that
A6: for m being Element of NAT st n2 <= m holds
|.((s9 . m) - (lim s9)).| < p / 2 by A2, A4, Def5;
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 + s9) . m) - ((lim s) + (lim s9))).| < p
let m be Element of NAT ; :: thesis: ( n <= m implies |.(((s + s9) . m) - ((lim s) + (lim s9))).| < p )
assume A7: n <= m ; :: thesis: |.(((s + s9) . m) - ((lim s) + (lim s9))).| < p
n2 <= n by XXREAL_0:25;
then n + n2 <= n + m by A7, XREAL_1:9;
then n2 <= m by XREAL_1:8;
then A8: |.((s9 . m) - (lim s9)).| < p / 2 by A6;
A9: |.(((s + s9) . m) - ((lim s) + (lim s9))).| = |.(((s . m) + (s9 . m)) - ((lim s) + (lim s9))).| by VALUED_1:1
.= |.(((s . m) - (lim s)) + ((s9 . m) - (lim s9))).| ;
n1 <= n by XXREAL_0:25;
then n + n1 <= n + m by A7, XREAL_1:9;
then n1 <= m by XREAL_1:8;
then |.((s . m) - (lim s)).| < p / 2 by A5;
then |.((s . m) - (lim s)).| + |.((s9 . m) - (lim s9)).| < (p / 2) + (p / 2) by A8, XREAL_1:10;
then (|.((s . m) - (lim s)).| + |.((s9 . m) - (lim s9)).|) + |.(((s + s9) . m) - ((lim s) + (lim s9))).| < p + (|.((s . m) - (lim s)).| + |.((s9 . m) - (lim s9)).|) by A9, COMPLEX1:142, XREAL_1:10;
hence |.(((s + s9) . m) - ((lim s) + (lim s9))).| < p by XREAL_1:8; :: thesis: verum

end;

s + s9 is convergent by A1, A2, Th13;
hence lim (s + s9) = (lim s) + (lim s9) by A3, Def5; :: thesis: verum