Proving that a series diverges Let $(a_n)$ positive sequence where $$\dfrac{n-1}{n} \leq \dfrac{a_{n+1}}{a_n} \leq \dfrac {n}{n+1}$$
Show that $\sum_{n=1}^{\infty} a_n$ diverges.
I already found out that 
$\sum_{n=1}^{\infty} (-1)^na_n$ is convergent and that $\lim a_n =0$.
I'm not sure how to approach this one.

For all $n \ge 1$,
$$a_{n+1} \ge \frac{n-1}{n}a_n \ge \frac{(n-1)(n-2)}{n(n-1)}a_{n-1}\ge \cdots \ge \frac{(n-1)(n-2)\cdots 2\cdot 1}{n(n-1)\cdots 3\cdot 2}a_2 = \frac{1}{n}a_2$$
Since $\sum \frac{1}{n}a_2$ diverges, by comparison the series $\sum a_n$ diverges.
