Question:
Find a sequence $\displaystyle \{a_n\}_{n=1}^{\infty}$ such that $a_n\rightarrow 0$ and $n\left|a_{n+1}-a_n\right|\rightarrow \infty$. If no such sequence exists, prove it.
My try:
At first I tried to find such sequence, but it seems to me that such sequence exists.
Next, I used the fact that if $a_n$ converges to some finite limit, then $\displaystyle \lim_{n\to\infty}a_{n+1}-a_n=0$, but it gets me to a dead end and I don't understand how I should tackle it.
Please help, thank you.
Take $$a_n=\frac{(-1)^n}{\sqrt{n}}.$$
Then $$ \lvert a_{n+1}-a_n\rvert=\left|\frac{(-1)^{n+1}}{\sqrt{n+1}}-\frac{(-1)^n}{\sqrt{n}}\right|= \left|\frac{1}{\sqrt{n+1}}+\frac{1}{\sqrt{n}}\right|\ge\frac{1}{\sqrt{n}}, $$ and hence $$ n\lvert a_{n+1}-a_n\rvert\ge\sqrt{n}\to\infty. $$