reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem Th23:
  for s being Real_Sequence, a st a > 0 &
   (for n st n>=1 holds s.n = n-root a)
  holds s is convergent & lim s = 1
proof
  let s be Real_Sequence, a;
  assume that
A1: a > 0 and
A2: for n st n>=1 holds s.n = n-root a;
 now
    let n be Nat;
    assume
A3: n>=1;
    hence s.n = n-root a by A2
      .= n -Root a by A1,A3,Def1;
  end;
  hence thesis by A1,PREPOWER:33;
end;
