reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th18:
  for X be RealNormSpace for seq be sequence of X holds seq is
constant implies ( seq is convergent & for k be Element of NAT holds lim seq =
  seq.k )
proof
  let X be RealNormSpace;
  let seq be sequence of X;
  assume
A1: seq is constant;
  then consider r be Point of X such that
A2: for n be Nat holds seq.n=r by VALUED_0:def 18;
  thus
A3: seq is convergent by A1,LOPBAN_3:12;
  now
    let k be Element of NAT;
    now
      let p be Real such that
A4:   0<p;
       reconsider n=0 as Nat;
      take n;
      let m be Nat such that
      n<=m;
      ||.(seq.m)-(seq.k).||=||.r-(seq.k).|| by A2
        .=||.r-r.|| by A2
        .=||.0.X.|| by RLVECT_1:15
        .=0 by NORMSP_1:1;
      hence ||.(seq.m)-(seq.k).||<p by A4;
    end;
    hence lim seq = seq.k by A3,NORMSP_1:def 7;
  end;
  hence thesis;
end;
