$\sum a_n$ and $\sum b_n$ converges. Prove $\sum a_nb_n$ converges.

Question

Let ${a_n}$ and ${b_n}$ sequences with positive terms, such that $\sum a_n$ and $\sum b_n$ converges. Prove that $\sum a_nb_n$ converges as well.

What I did: $\sum a_nb_n \le \left(\sum a_n\right) \left(\sum b_n\right)$ and by comparison test we are done.

However, when I asked my teacher regarding that solution, he told me that I can assume that inequality for finite numbers but for infinite numbers, it is not valid. It seems to me that $\left(\sum a_n\right) \left(\sum b_n\right)$ is absolute converges so that are no Riemann issues with this.

Can any one enlighten me and explain why is this invalid?

Answer 1

The way you started and since $\left(a_n\right)_{n\in\mathbb{N}}$ and $\left(b_n\right)_{n\in\mathbb{N}}$ are positive then for partial sums we have

$$0\leq\sum\limits_{n=1}^{k} a_nb_n\leq \left(\sum\limits_{n=1}^{k} a_n\right)\left(\sum\limits_{n=1}^{k} b_n\right)$$ The limit has the property to preserve the inequality (number 7 down the list), then $$0\leq\lim\limits_{k\rightarrow\infty}\sum\limits_{n=1}^{k} a_nb_n\leq \lim\limits_{k\rightarrow\infty}\left(\sum\limits_{n=1}^{k} a_n\right)\left(\sum\limits_{n=1}^{k} b_n\right) =\lim\limits_{k\rightarrow\infty} \left(\sum\limits_{n=1}^{k} a_n\right)\cdot \lim\limits_{k\rightarrow\infty} \left(\sum\limits_{n=1}^{k} b_n\right) $$ which is

$$0\leq \sum\limits_{n=1} a_nb_n\leq \left(\sum\limits_{n=1} a_n\right)\cdot \left(\sum\limits_{n=1}b_n\right) $$

Since we know that $0\leq\sum\limits_{n=1}a_n < \infty$ and $0\leq\sum\limits_{n=1}b_n < \infty$, thus we have $0\leq \sum\limits_{n=1} a_nb_n <\infty$

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Another way, since $0\leq\sum\limits_{n=1}b_n < \infty$ then $\lim\limits_{n\rightarrow\infty}b_n=0$ which also means $\left(b_n\right)_{n\in\mathbb{N}}$ is bounded or $0\leq b_n < M, \forall n\in\mathbb{N}$. Then $$0\leq\sum\limits_{n=1}^{k} a_nb_n\leq M \left(\sum\limits_{n=1}^{k} a_n\right)$$ Taking the limit $$0\leq\sum\limits_{n=1} a_nb_n\leq M \left(\sum\limits_{n=1} a_n\right)<\infty$$

Answer 2

Since $\sum\limits_{k = 1}^\infty b_k$ converges, then $b_n \to 0 \ (n \to \infty)$. Note that $\{b_n\}$ is a sequence of positive numbers, thus there exists $N \in \mathbb{N}_+$ such that$$ 0 < b_n < 1. \quad \forall n > N $$ Therefore,$$ 0 < \sum_{k = 1}^\infty a_k b_k = \sum_{k = 1}^N a_k b_k + \sum_{k = N + 1}^\infty a_k b_k \leqslant \sum_{k = 1}^N a_k b_k + \sum_{k = N + 1}^\infty a_k < +\infty. $$

Answer 3

Basically you can use this fact

For sure, if $x_n>0$ and $\sum x_n$ converges then exists $n_0\in\Bbb N$ such that $n>n_0\implies x_n<1\implies x_n^2<x_n$. Then by comparison $\sum x_n^2$ converges.

If $a_n,b_n>0$ are such $\sum a_n$ and $\sum b_n$ converges, then $\sum a_n^2$ and $\sum b_n^2$ converges as well. Then, $$(a_n-b_n)^2\geq0\implies a_n^2+b_n^2\geq2a_nb_n,\quad\forall n\in\Bbb N.$$ Again, by comparison, have that $\sum a_nb_n$ converges.