I have two series $\sum_{k=1}^\infty x_k$ and $\sum_{k=1}^\infty y_k$ and I know both converge, and further one of them converges absolutely. Then I'd like to prove that $\sum_{k=1}^\infty x_ky_k$ converges as well.

I was thinking:

With out loss of generality suppose $\sum_{k=1}^\infty |x_k| = L$, for some real number $L$. Since $\sum_{k=1}^\infty y_k$ converges then $\lim_{k \to \infty} |y_k| = 0$, so for large $k$, we have $|x_ky_k| < |x_k|$, so if $\sum_{k=1}^\infty |x_k|$ by comparison so must $\sum_{k=1}^\infty |x_ky_k|$, which implies the convergence of $\sum_{k=1}^\infty x_ky_k$.

My worry is the inequality only holds for large $k$ so is that sufficient? Also, at large $k$ doesn't $|x_k| \to 0$ as well? Essentially the crux of my argument is using comparison test and I'm having doubts on the inequality.


Yes, this is correct. Suppose the inequality holds for all $k>N$. Then the sum of the first $N$ terms is a number and the comparison test works for the rest of the series.


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