How can I find $\lim_{n\to \infty} a_n$ 
Let $$a_n=\left(1-\dfrac{1}{\sqrt2}\right)\dots \left(1-\dfrac{1}{\sqrt{n+1}}\right),n\ge1$$ Then find $\lim_{n\to \infty} a_n$.

How can I proceed? I am stuck at the first step. Please help.

Note that $a_n>0$ and
$$\lim_{n\to\infty}\log a_n=\sum_{k=1}^\infty \log  \left(1-\dfrac{1}{\sqrt{k+1}}\right)$$
is divergent to $-\infty$  by comparaison test with Riemann divergent series:
$$\log  \left(1-\dfrac{1}{\sqrt{k+1}}\right)\sim_\infty -\dfrac{1}{\sqrt{k}}$$
hence
$$\lim_{n\to\infty} a_n=0$$
