Does $\displaystyle \{a_n\} $ converging imply $\displaystyle \sum_{n=1}^\infty (-1)^na_n $ is bounded?
What chip said! The partial sums you give are bounded.
To prove it, let $\displaystyle b_n=a_n-a$ where $\displaystyle a=\lim_{n\to \infty}a_n$. By the alternating series test, $\displaystyle \sum_{j=1}^\infty (-1)^jb_j$ converges, which immediately implies the partial sums $\displaystyle \sum_{j=1}^m(-1)^ja_j$ are bounded.