The following problem comes from Walter’s Rudin classic text Real and Complex Analysis. It’s Exercise 7 of Chapter 4 (Elementary Hilbert Space Theory).
Suppose is a sequence of positive numbers such that for any nonnegative sequence . Then .
Proof. Suppose that . By induction, we can construct a countable collection of disjoint finite sets which partition and satisfy
We will construct a nonnegative sequence such that , but . For each , define
and for , define . Since the partial sums of the series are nonnegative, the order of summation is irrelevant, provided the series converges. Thus,
since , and
We arrive at a contradiction, which completes the proof.