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.

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