reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th17:
  (for n holds 0<=s.n) implies (Partial_Sums(s) is bounded_above
  iff s is summable)
proof
  assume for n holds 0<=s.n;
  then Partial_Sums(s) is non-decreasing by Th16;
  hence Partial_Sums(s) is bounded_above implies s is summable;
  assume s is summable;
  then Partial_Sums(s) is convergent;
  then Partial_Sums(s) is bounded;
  hence thesis;
end;
