Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$
Hint
Prove it by induction and you should show in the inductive step this inequality:
$$\frac{2n+1}{2n+2}\frac{1}{\sqrt{3n+1}}\le \frac{1}{\sqrt{3n+4}}$$ which is simple to see it by taking the square.
Added: Notice that $$(2n+2)^2(3n+1)-(2n+1)^2(3n+4)=n\ge0$$