I was wondering if my proof for the following problem is correct:


If $\displaystyle \sum_{n} a_n$ converges, and if $\{b_n\}$ is monotonic and bounded, prove that $\displaystyle \sum_{n} a_n b_n$ converges.


Assume $\displaystyle \sum_{n} a_n$ converges and $\{b_n\}$ is monotonic and bounded. Then the partial sums of $a_n$ form a bounded sequence; that is, $\displaystyle A_k = \sum_{n=1}^{k} a_n$ is a bounded sequence.

Theorem 3.14 states that:

Suppose $\{s_n\}$ is monotonic. Then $\{s_n\}$ converges if and only if it is bounded.

Thus, by Theorem 3.14, $\{b_n\}$ converges. Let $\displaystyle L = \lim_{n \rightarrow \infty} b_n$. Consider the sequence $\{b_n - L\}$. Then $\{b_n - L\}$ is also a monotonic and bounded sequence, so by Theorem 3.14, $\displaystyle \lim_{n \rightarrow \infty} b_n - L$ converges and $\displaystyle \lim_{n \rightarrow \infty} b_n - L = 0$.

Theorem 3.42 states that:

Suppose: (a) the partial sums $A_n$ of $\displaystyle \sum_{n} a_n$ form a bounded sequence; (b) $b_0 \geq b_1 \geq b_2 \geq \ldots$ (c) $\displaystyle \lim_{n \rightarrow \infty} b_n = 0$ Then $\displaystyle \sum_{n} a_n b_n$ converges.

Thus, by Theorem 3.42, $\displaystyle \sum_{n} a_n (b_n - L)$ converges, so $\displaystyle \sum_{n} a_n b_n - \sum_{n} a_n L$ converges. In particular, $\displaystyle \sum_{n} a_n b_n$ converges. $\blacksquare$

Thank you.


Just one comment: In your (almost) last line, you split up the infinite sum without knowing that each of the pieces exist. You should conclude that $\sum a_{n}b_{n}$ converges as follows:

$$ \begin{aligned} \sum a_{n}b_{n}&=\sum [a_{n}(b_{n}-L)+La_{n}]=\sum a_{n}(b_{n}-L)+L\sum a_{n}<\infty \end{aligned} $$

since $\sum a_{n}<\infty$.

  • $\begingroup$ Thank you, I didn't know that when you split up infinite sums, you need to justify why each sum exists. $\endgroup$ – Frederic Chopin Nov 12 '17 at 22:15
  • $\begingroup$ Think $\sum(1/n-1/n)=0\not=\sum (1/n)-\sum (1/n)$. $\endgroup$ – ervx Nov 12 '17 at 22:16
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    $\begingroup$ Also, I just realized something. I didn't prove that $\{b_n - L\}$ is a monotonically decreasing sequence, as it should be in Theorem 3.42. All is good if $\{b_n\}$ is monotonically decreasing, but what happens if $\{b_n\}$ is monotonically increasing? Then it is true that $\{b_n - L\}$ is a monotonically decreasing sequence. $\endgroup$ – Frederic Chopin Nov 12 '17 at 22:17
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    $\begingroup$ You can consider $L-b_{n}$ instead, in this case. $\endgroup$ – ervx Nov 12 '17 at 22:18
  • $\begingroup$ Oh yes! Good point for both $\endgroup$ – Frederic Chopin Nov 12 '17 at 22:32

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