reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem
  seq is convergent & (ex k st for n st n >= k holds seq9.n = seq.n)
  implies lim seq = lim seq9
proof
  assume that
A1: seq is convergent and
A2: ex k st for n st n >= k holds seq9.n = seq.n;
  consider k such that
A3: for n st n >= k holds seq9.n = seq.n by A2;
A4: now
    let r;
    assume r > 0;
    then consider m1 such that
A5: for n st n >= m1 holds dist((seq.n) , (lim seq)) < r by A1,Def2;
A6: now
      assume
A7:   m1 <= k;
      take m = k;
      let n;
      assume
A8:   n >= m;
      then n >= m1 by A7,XXREAL_0:2;
      then dist((seq.n) , (lim seq)) < r by A5;
      hence dist((seq9.n) , (lim seq)) < r by A3,A8;
    end;
    now
      assume
A9:   k <= m1;
      take m = m1;
      let n;
      assume
A10:  n >= m;
      then seq9.n = seq.n by A3,A9,XXREAL_0:2;
      hence dist((seq9.n) , (lim seq)) < r by A5,A10;
    end;
    hence ex m st for n st n >= m holds dist((seq9.n) , (lim seq)) < r by A6;
  end;
  seq9 is convergent by A1,A2,Th2;
  hence thesis by A4,Def2;
end;
