
theorem Th10:
  for X be RealNormSpace for seq,seq1 be sequence of X holds seq
  is convergent & (ex k be Nat st seq=seq1 ^\k) implies seq1 is
  convergent
proof
  let X be RealNormSpace;
  let seq,seq1 be sequence of X;
  assume that
A1: seq is convergent and
A2: ex k be Nat st seq=seq1 ^\k;
  consider k be Nat such that
A3: seq=seq1 ^\k by A2;
  consider g1 be Element of X such that
A4: for p be Real st 0<p ex n be Nat st
   for m be Nat st n<=m holds ||.seq.m-g1.||<p by A1;
  take g1;
  let p be Real;
  assume 0<p;
  then consider n1 be Nat such that
A5: for m be Nat st n1<=m holds ||.seq.m-g1.||<p by A4;
  take n=n1+k;
  let m be Nat;
  assume
A6: n<=m;
  then consider l be Nat such that
A7: m=n1+k+l by NAT_1:10;
  reconsider l as Nat;
  m-k=n1+l+k+-k by A7;
  then reconsider m1=m-k as Nat;
  now
    assume not n1<=m1;
    then m1+k<n1+k by XREAL_1:6;
    hence contradiction by A6;
  end;
  then m1+k=m & ||.seq.m1-g1.||<p by A5;
  hence thesis by A3,NAT_1:def 3;
end;
