reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem Th16:
  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 g1 such that
A3: for p st 0<p ex n st for m st n<=m holds |.seq.m-g1.|<p by A2;
  take g1;
  let p;
  assume 0<p;
  then consider n1 such that
A4: for m st n1<=m holds |.seq.m-g1.|<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 thesis by A4,A7;
end;
