 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem
  Rseq is nonnegative-yielding implies
   (Partial_Sums Rseq is P-convergent
  iff Partial_Sums Rseq is bounded_below bounded_above)
proof
   assume Rseq is nonnegative-yielding; then
a2:Partial_Sums Rseq is non-decreasing by th80a;
   hence Partial_Sums Rseq is P-convergent
     implies Partial_Sums Rseq is bounded_below bounded_above;
   assume Partial_Sums Rseq is bounded_below bounded_above;
   hence Partial_Sums Rseq is P-convergent by a2;
end;
