Proof that a sequence is bounded by inspection of series. If $\sum\limits c_n$  converges absolutely and $\sum\limits c_nk_n $ also converges absolutely, then $k_n$ must be bounded. How do I prove this? I think this can be proven by contradiction. Anyone?

The correct statement might be:

A sequence $(k_n)$ is bounded if and only if, for every absolutely convergent series $\sum\limits c_n$, the series $\sum\limits c_nk_n $ converges absolutely.

If $(k_n)$ is bounded, the conclusion holds. If $(k_n)$ is unbounded, for every integer $i\geqslant1$, choose $n(i)$ such that $|k_{n(i)}|\geqslant i$ and assume the integers $n(i)$ are all different (can you see why such a sequence $(n(i))$ always exists?). Define $(c_n)$ by $c_{n(i)}=1/i^2$ and $c_n=0$ if $n$ is not one of the $n(i)$s. Then the series $\sum\limits c_n$ is absolutely convergent because $\sum1/i^2$ is, while the series $\sum\limits c_nk_n $ is not because $\sum1/i$ is not.
