
theorem
  for seq1,seq2 be ExtREAL_sequence
    st seq2 is subsequence of seq1 & seq1 is convergent_to_finite_number
   holds seq2 is convergent_to_finite_number & lim seq1 = lim seq2
proof
   let seq1,seq2 be ExtREAL_sequence;
   assume that
A1: seq2 is subsequence of seq1 and
A2: seq1 is convergent_to_finite_number;
   not seq1 is convergent_to_+infty & not seq1 is convergent_to_-infty
     by A2,MESFUNC5:50,51; then
   consider g be Real such that
B3: lim seq1 = g & (for p be Real st 0<p
     ex n be Nat st for m be Nat st n<=m holds |.seq1.m-lim seq1.| < p)
  & seq1 is convergent_to_finite_number by A2,MESFUNC5:def 12;
   reconsider LIM2 = lim seq1 as R_eal;
   ex g be Real st
   for p be Real st 0<p ex n be Nat st for m be Nat st n<=m
    holds |.seq2.m-g qua ExtReal.| < p
   proof
    take g;
    hereby let p be Real;
     assume 0<p;
     then consider n1 be Nat such that
A4:   for m be Nat st n1<=m holds |.seq1.m-g qua ExtReal.|<p by B3;
     take n=n1;
     let m be Nat such that
A5:   n<=m;
     consider Nseq be increasing sequence of NAT such that
A6:   seq2=seq1*Nseq by A1,VALUED_0:def 17;
     m<=Nseq.m by SEQM_3:14; then
A7:  n<=Nseq.m by A5,XXREAL_0:2;
     seq2.m=seq1.(Nseq.m) by A6,FUNCT_2:15,ORDINAL1:def 12;
     hence |.seq2.m - g qua ExtReal .| < p by A4,A7;
    end;
   end;
   hence
A8: seq2 is convergent_to_finite_number by MESFUNC5:def 8;
   for p be Real st 0<p
        ex n be Nat st for m be Nat st n<=m holds |.seq2.m-LIM2.| < p
   proof
    let p be Real;
    assume 0<p; then
    consider n1 be Nat such that
A10: for m be Nat st n1<=m holds |.seq1.m-lim seq1.|<p by B3;
    take n=n1;
    let m be Nat such that
A11: n<=m;
    consider Nseq be increasing sequence of NAT such that
A12: seq2=seq1*Nseq by A1,VALUED_0:def 17;
    m<=Nseq.m by SEQM_3:14; then
A13:n<=Nseq.m by A11,XXREAL_0:2;
    seq2.m=seq1.(Nseq.m) by A12,FUNCT_2:15,ORDINAL1:def 12;
    hence |.seq2.m - LIM2 .| < p by A10,A13;
   end;
   hence thesis by A8,B3,MESFUNC5:def 12;
end;
