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 Th8:
  X is compact implies X is closed
proof
  assume
A1: X is compact;
  assume not X is closed;
  then consider s1 such that
A2: rng s1 c= X and
A3: s1 is convergent & not lim s1 in X;
  ex s2 st s2 is subsequence of s1 & s2 is convergent & (lim s2) in X by A1,A2;
  hence contradiction by A3,SEQ_4:17;
end;
