TITLE: Convergence of a sequence $c_n$
QUESTION [10 upvotes]: Suppose that $(a_n)$ and $(b_n)$ be sequences such that $\lim (a_n)=0$ and $\displaystyle \lim \left( \sum_{i=1}^n b_i \right)$ exists. Define $c_n = a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1$. Prove that $\lim (c_n)=0$

REPLY [2 votes]: Suppose $a_n\to0$ monotonically.
Let $B_k=\sum\limits_{j=k}^\infty b_j$. Since the sum of $b_j$ converges, $\bar{B}_n=\sup\limits_{k\ge n}\left|B_k\right|$ decreases monotonically to $0$.
Let $\epsilon\gt0$. Choose $N$ so that if $n\ge N$, $a_{n/2}\bar{B}_0\le\epsilon/4, a_0\bar{B}_{n/2}\le\epsilon/4$
$$
\begin{align}
|c_n|&
=\left|\sum_{k=0}^na_kb_{n-k}\right|\\
&=\left|\sum_{k=0}^na_k(B_{n-k}-B_{n-k+1})\right|\\
&=\left|a_nB_0-a_0B_{n+1}+\sum_{k=0}^{n-1}a_kB_{n-k}-\sum_{k=0}^{n-1}a_{k+1}B_{n-k}\right|\\
&=\left|a_nB_0-a_0B_{n+1}+\sum_{k=0}^{n-1}(a_k-a_{k+1})B_{n-k}\right|\\
&=\left|a_n\bar{B}_0-a_0\bar{B}_n+\sum_{k=0}^{n/2-1}(a_k-a_{k+1})B_{n-k}+\sum_{k=n/2}^{n-1}(a_k-a_{k+1})B_{n-k}\right|\\
&\le|a_n\bar{B}_0|+|a_0\bar{B}_n|+|a_0\bar{B}_{n/2}|+|a_{n/2}\bar{B}_0|\\
&\le\epsilon
\end{align}
$$
Thus, $c_n\to0$.