
theorem Th12:
  for X be ComplexNormSpace, seq be sequence of X holds seq is
  constant implies seq is convergent
proof
  let X be ComplexNormSpace;
  let seq be sequence of X;
  assume seq is constant;
  then consider r be Element of X such that
A1: for n be Nat holds seq.n=r by VALUED_0:def 18;
  take g=r;
  let p be Real such that
A2: 0<p;
  take n=0;
  let m be Nat such that
  n<=m;
  ||.(seq.m)-g.||=||.r-g.|| by A1
    .=||.0.X.|| by RLVECT_1:15
    .=0;
  hence thesis by A2;
end;
