reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem Th17:
  seq1 is subsequence of seq & seq is convergent implies seq1 is convergent
proof
  assume that
A1: seq1 is subsequence of seq and
A2: seq is convergent;
  consider g being Complex such that
A3: for p st 0<p ex n st for m st n<=m holds |.(seq.m)-g.|<p by A2;
  take t=g;
  let p;
  assume 0<p;
  then consider n1 such that
A4: for m st n1<=m holds |.(seq.m)-g.|<p by A3;
  take n=n1;
  let m such that
A5: n<=m;
  consider Nseq such that
A6: seq1=seq*Nseq by A1,VALUED_0:def 17;
  m<=Nseq.m by SEQM_3:14;
  then
A7: n<=Nseq.m by A5,XXREAL_0:2;
   m in NAT by ORDINAL1:def 12;
  then seq1.m=seq.(Nseq.m) by A6,FUNCT_2:15;
  hence |.(seq1.m)-t.|<p by A4,A7;
end;
