Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.

Question

Problem: prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction. I tried some, but stopped in $\sqrt[2^n]{n+1}$. Also tried with $2\sqrt{3\cdots}<3^2$ and so on.