
theorem
  for seq1,seq2 be ExtREAL_sequence
   st seq2 is subsequence of seq1 & seq1 is convergent_to_-infty
    holds seq2 is convergent_to_-infty & lim seq2=-infty
proof
   let seq1,seq2 be ExtREAL_sequence;
   assume that
A1: seq2 is subsequence of seq1 and
A2: seq1 is convergent_to_-infty;
   now let g be Real;
    assume g < 0; then
    consider n1 be Nat such that
A4:  for m be Nat st n1<=m holds seq1.m <= g by A2,MESFUNC5:def 10;
    take n = n1;
    consider Nseq be increasing sequence of NAT such that
A5:  seq2 = seq1 * Nseq by A1,VALUED_0:def 17;
    let m be Nat;
    assume A6: n <= m;
    m <= Nseq.m by SEQM_3:14; then
A7: n <= Nseq.m by A6,XXREAL_0:2;
    seq2.m = seq1.(Nseq.m) by A5,FUNCT_2:15,ORDINAL1:def 12;
    hence seq2.m <= g by A4,A7;
   end;
   hence seq2 is convergent_to_-infty by MESFUNC5:def 10;
   hence thesis by MESFUNC5:def 12;
end;
