 reserve n,i for Nat;

theorem
  for f being nonnegative-yielding Real_Sequence st f is summable holds
    for p being Real st p > 0 holds
    ex i being Element of NAT st Sum (f ^\ i) < p
  proof
    let f be nonnegative-yielding Real_Sequence;
    assume
A1: f is summable;
    let p be Real;
    assume
B1: p > 0;
    assume
O1: for i being Element of NAT holds Sum (f ^\ i) >= p;
    set S = Sum f;
    Partial_Sums f is convergent by A1,SERIES_1:def 2; then
    consider n being Nat such that
D1: for m being Nat st n <= m holds
      |. (Partial_Sums f.m - S) .| < p by B1,SEQ_2:def 7;
    reconsider m = n + 1 as Element of NAT;
    reconsider m1 = m + 1 as Element of NAT;
x1: Partial_Sums f.m + Sum (f ^\ m1) = S by A1,SERIES_1:15;
    set R = f ^\ m1;
Y1: R is summable by A1,SERIES_1:12;
    for n being Nat holds 0 <= R.n by RINFSUP1:def 3;
    then Sum (f ^\ m1) >= 0 by Y1,SERIES_1:18; then
    -(S - Partial_Sums f.m) <= -0 by x1; then
    per cases;
    suppose
X3:   Partial_Sums f.m - S < 0;
      |. (Partial_Sums f.m - S) .| < p by D1,NAT_1:11; then
      -(Partial_Sums f.m - S) < p by X3,ABSVALUE:def 1;
      hence thesis by O1,x1;
    end;
    suppose
      Partial_Sums f.m - S = 0;
      hence thesis by O1,x1,B1;
    end;
  end;
