(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose we have a sequence {x} = {x_1, x_2, ...} and we know that [itex]\{x\}\in\ell^2[/itex], i.e. [itex]\sum^\infty x^2_n<\infty[/itex]. Does it follow that there exists a K>0 such that [itex]x_n<K/n[/itex] for all n?

2. Relevant equations

The converse is easy, [itex]\sum 1/n^2 = \pi^2/6[/itex], so there would be a finite upper bound for [itex]\sum^\infty x^2_n<\infty[/itex].

3. The attempt at a solution

I'm stuck. I cannot think of a counterexample and my hunch is that it's true. I was hoping that maybe from [itex]\sum^\infty x^2_n=L[/itex] for some L I could derive a bound for the elements of the sum involving the summation index, but to no avail so far.

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# If sequence {x} is in l2, does x_n<k/n follow?

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