
theorem Th4:
  for X be ComplexNormSpace, seq be sequence of X holds seq is
  convergent implies
  for s be Real st 0 < s ex n be Nat st for m be
  Nat st n<=m holds ||.seq.m -seq.n.||<s
proof
  let X be ComplexNormSpace;
  let seq be sequence of X;
  assume seq is convergent;
  then consider g1 be Element of X such that
A1: for s be Real st 0<s ex n be Nat st
    for m be Nat st n<=m holds ||.seq.m -g1.||<s;
  let s be Real;
  assume 0 < s;
  then consider n be Nat such that
A2: for m be Nat st n<=m holds ||.seq.m -g1.||<s/2 by A1;
  now
    let m be Nat;
    assume n<=m;
    then
A3: ||.seq.m -g1.||<s/2 by A2;
A4: ||.seq.m -g1+(g1-seq.n).||<=||.seq.m -g1.||+||.g1-seq.n.|| & ||.seq.m
    -g1+( g1-seq.n).|| = ||.seq.m -seq.n.|| by Th3,CLVECT_1:def 13;
    ||.seq.n -g1.||<s/2 by A2;
    then ||.g1-seq.n.||<s/2 by CLVECT_1:108;
    then ||.seq.m -g1.||+||.g1-seq.n.||<s/2+s/2 by A3,XREAL_1:8;
    hence ||.seq.m -seq.n.|| < s by A4,XXREAL_0:2;
  end;
  hence thesis;
end;
