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\noindent {\bf Question}

\noindent In each of the following, $\sum_{n=1}^\infty a_n$ is a
convergent series with positive terms.
\begin{enumerate}
\item Prove that, if $\{ c_n\}$ is a sequence of positive terms
satisfying $\lim_{n\rightarrow\infty} c_n =0$, then
$\sum_{n=1}^\infty a_n c_n$ converges;
\item Prove that, if $\{ c_n\}$ is a sequence of positive terms
satisfying $\lim_{n\rightarrow\infty} c_n =c\neq 0$, then
$\sum_{n=1}^\infty a_n c_n$ converges.
\end{enumerate}
\medskip

\noindent {\bf Answer}

\noindent Let $S_k =\sum_{n=1}^k a_n$ be the $k^{th}$ partial sum
of $\sum_{n=1}^\infty a_n$.
\begin{enumerate}
\item Since $\lim_{n\rightarrow\infty} c_n =0$ and $c_n >0$ for all
$n$, there exists $M >0$ so that $0 < c_n <1$ for $n >M$.  Let $M
={\rm max}(1, c_1, c_2, \ldots, c_M)$, and note that $c_n\le M$
for all $n\ge 1$.  In particular, the $k^{th}$ partial sum of the
series $\sum_{n=1}^\infty a_n c_n$ satisfies
\[ \sum_{n=1}^k a_n c_n \le \sum_{n=1}^k a_n M = M\: S_k. \]
Hence, the sequence of partial sums of the series
$\sum_{n=1}^\infty a_n c_n$ forms a monotonic (since all the $a_n$
and $c_n$ are positive), bounded (above by $M\sum_{n=1}^\infty
a_n$, below by $0$) sequence, and so converges.  That is,
$\sum_{n=1}^\infty a_n c_n$ converges.
\item The proof in the case that $\lim_{n\rightarrow\infty} c_n =c\ne
0$ is very similar to the proof in the case that
$\lim_{n\rightarrow\infty} c_n =0$.  Since
$\lim_{n\rightarrow\infty} c_n =c\ne 0$, there exists $M >0$ so
that $c_n <c +1$ for $n >M$. Let $M ={\rm max}(c+1, c_1, c_2,
\ldots, c_M)$, and note that $c_n\le M$ for all $n\ge 1$.  In
particular, the $k^{th}$ partial sum of the series
$\sum_{n=1}^\infty a_n c_n$ satisfies
\[ \sum_{n=1}^k a_n c_n \le \sum_{n=1}^k a_n M = M\: S_k. \]
Hence, the sequence of partial sums of the series
$\sum_{n=1}^\infty a_n c_n$ forms a monotonic (since all the $a_n$
and $c_n$ are positive), bounded (above by $M\sum_{n=1}^\infty
a_n$, below by $0$) sequence, and so converges.  That is,
$\sum_{n=1}^\infty a_n c_n$ converges.
\end{enumerate}


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