A nice finite induction problem: $\sqrt{2 \sqrt{3 \sqrt{4 \ldots \sqrt{n}}}}<3.$

Question

for all $n\in\mathbb{N}$, $$\sqrt{2 \sqrt{3 \sqrt{4 \ldots \sqrt{n}}}}<3.$$ Any help proving this problem by finite induction. I tried to use $\sqrt{2}\cdot\sqrt[4]{3}\cdot\sqrt[8]{4}\cdots\sqrt[2^{n-1}]{n}<3$ as a hypothesis, but I was not successful. I'm trying to resolve this issue in my article, which brings together questions like this.

Answer 1

Hint: Let $n$ be a positive integer. Then $$\sqrt{2 \sqrt{3 \sqrt{4 \ldots \sqrt{n}}}}=\prod_{k=2}^n\sqrt[2^{k-1}]{k}=\prod_{k=2}^n\exp({2^{1-k}\ln{k}})=\exp\left(\sum_{k=2}^n\frac{\ln{k}}{2^{k-1}}\right).$$ Can you find an upper bound for that sum?