reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th30:
  dseq is convergent & lim(dseq)=Sum(eseq)
proof
  for eps st eps>0 holds ex N st for n st n>=N holds dseq.n>Sum(eseq)-eps
  proof
    let eps;
    assume
A1: eps>0;
    then consider K such that
A2: Partial_Sums(eseq).K>Sum(eseq)-eps/2 by Th23,Th27,XREAL_1:139;
A3: Partial_Sums(eseq).K-eps/2>Sum(eseq)-eps/2-eps/2 by A2,XREAL_1:9;
    deffunc F(Nat)=Partial_Sums(cseq($1)).K;
    consider dseqK being Real_Sequence such that
A4: for n holds dseqK.n=F(n) from SEQ_1:sch 1;
    dseqK is convergent & lim(dseqK)=Partial_Sums(eseq).K by A4,Th24;
    then consider N such that
A5: for n st n>=N holds dseqK.n>Partial_Sums(eseq).K-eps/2 by A1,Th25,
XREAL_1:139;
    take N1 = N+1;
    let n;
    assume
A6: n>=N1;
    then ( for k holds cseq(n).k>=0)& cseq(n) is summable by Th14,Th21;
    then Sum(cseq(n))>=Partial_Sums(cseq n).K by Th29;
    then dseq.n>=Partial_Sums(cseq n).K by A6,Th21;
    then
A7: dseq.n>=dseqK.n by A4;
    N+1>=N+0 by XREAL_1:6;
    then n>=N by A6,XXREAL_0:2;
    then dseqK.n>Partial_Sums(eseq).K-eps/2 by A5;
    then dseq.n>Partial_Sums(eseq).K-eps/2 by A7,XXREAL_0:2;
    hence thesis by A3,XXREAL_0:2;
  end;
  hence thesis by Th26,Th28;
end;
