reserve n,n1,m,k for Nat;
reserve x,y for set;
reserve s,g,g1,g2,r,p,p2,q,t for Real;
reserve s1,s2,s3 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve X for Subset of REAL;

theorem
  [.s,g.] is compact
proof
  for s1 st rng s1 c= [.s,g.] ex s2 st s2 is subsequence of s1 & s2 is
  convergent & lim s2 in [.s,g.]
  proof
    let s1;
A1: [.s,g.] is closed by Th5;
    assume
A2: (rng s1) c= [.s,g.];
    then consider s2 be Real_Sequence such that
A3: s2 is subsequence of s1 and
A4: s2 is convergent by Th4,SEQ_4:40;
    take s2;
    ex Nseq st s2 = s1*Nseq by A3,VALUED_0:def 17;
    then rng s2 c= rng s1 by RELAT_1:26;
    then rng s2 c= [.s,g.] by A2;
    hence thesis by A3,A4,A1;
  end;
  hence thesis;
end;
