reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th52:
  seq is summable implies seq is convergent & lim seq = 0c
proof
  assume
A1: seq is summable;
  then Re Partial_Sums(seq) is convergent;
  then Partial_Sums(Re seq) is convergent by Th26;
  then
A2: Re seq is summable;
  Im Partial_Sums(seq) is convergent by A1;
  then Partial_Sums(Im seq) is convergent by Th26;
  then
A3: Im seq is summable;
  then lim(Im seq)=0 by SERIES_1:4;
  then
A4: Im(lim(seq))=0 by A2,A3,Th42;
  lim(Re seq)=0 by A2,SERIES_1:4;
  then Re(lim(seq))=0 by A2,A3,Th42;
  hence thesis by A2,A3,A4,Lm1,Th42,COMPLEX1:13;
end;
