# convergence with a bounded sequence and a sequence that tends to zero

How to prove that if $\sum k_n$ converges absolutely and $\lim_{n \to +\infty}c_n = 0$ (where $c_n$ is a sequence) then $\sum k_n c_n$ converges absolutely

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Hint: the implication holds as soon as the sequence $(c_n)$ is bounded. –  Did Sep 29 '13 at 18:32

Hint: If $\lim_{n\to\infty}c_n=0$, then for some $N\geq1$, we have $|c_k|\leq1$ for $k\geq N$. Now think of the comparison test.
it is just the inequality $\vert k_k \vert \vert c_k \vert \leq \vert k_k \vert$ for some $k \geq N$ now iam ready to compare right? –  Danny Sep 29 '13 at 1:49