Real Analysis: Limit of Sequence

There exists a sequence such that and there exists a sequence of non-negative real numbers such that .

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I'm having some trouble figuring out where to begin on this one. I've tried re-writing it in summation notation and manipulating that in hopes of being able to cancel y terms but hasn't gotten me anywhere. I thought it might be possible to rewrite the expression as a subsequence of and then take the fact that a subsequence converges to the same limit as the original sequence but I would not no where to begin to construct the subsequence.