Let $\{a\_n\}\_{n\geq 1}$ be a sequence of strictly decreasing positive numbers i.e., $$a\_1>a\_2>a\_3>\cdots$$

Then which of the following is always true?

* $\lim \_{n\rightarrow \infty} a\_n=0$
* $\lim \_{n\rightarrow \infty} \frac{a\_n}{n}=0$
* $\sum\_{n=1}^{\infty}\frac{a\_n}{n}$ converges.
* $\sum\_{n=1}^{\infty}\frac{a\_n}{n^2}$ converges.

I guess all options are correct but i could not give a concrete proof.

For third bullet, Take $b\_n=\frac{a\_n}{n}$ and look for ratio test :

$\lim\_{n\rightarrow \infty}\frac{b\_{n+1}}{b\_n}=\lim\_{n\rightarrow \infty}\frac{a\_{n+1}}{n+1}.\frac{n}{a\_n}\rightarrow 0$ Thus $\sum\_{n=1}^{\infty}\frac{a\_n}{n}$ converges.

For fourth bullet, I would just use :

As $a\_n$ are positive and $\sum\_{n=1}^{\infty}\frac{a\_n}{n}$ converges with $\frac{a\_n}{n^2}<\frac{a\_n}{n}$.

we would say $\sum\_{n=1}^{\infty}\frac{a\_n}{n^2}$ converges.

I can not convince myself that this is a proof.

So, please help me to make this a bit more clear.

Thank you :)