What is $\lim\limits_{n\to\infty}(\sqrt2-\sqrt[3]2)\cdots(\sqrt2-\sqrt[n]2)$? How to approach?

Question

$$\lim\limits_{n\to\infty}(\sqrt2-\sqrt[3]2)(\sqrt2-\sqrt[4]2)(\sqrt2-\sqrt[5]2)\cdots(\sqrt2-\sqrt[n]2)$$

Could you tell me how to approach this kind of question? How do I find the limit of this sequence?

I know that for very large $n$ the each bracket is more than $1$, so my guess is its going to infinity, how do I prove such a thing?

Answer 1

Formally, we can say that $\sqrt 2\gt\sqrt[3]2\gt\sqrt[4]2\gt\dots\gt\sqrt[n]2\gt 1$ as $n\to\infty$, and $0\lt\sqrt 2-\sqrt[3] 2\lt\sqrt 2-\sqrt[4] 2\lt\dots\lt\sqrt 2-\sqrt[n] 2\lt\sqrt 2-1\lt \frac 12,$ therefore

$$0\le\lim_{n\to\infty}\prod_{i=3}^n(\sqrt 2-\sqrt[i] 2)\le\lim_{n\to\infty}\left(\frac 12\right)^n=0$$