TITLE: Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$
QUESTION [6 upvotes]: I need to find: 

$$
\lim_{n \to \infty } (\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2}) 
$$

So far, I think that $0<\sqrt{2}-\sqrt[n]{2}<1$, and it seems to me that the limit will approach zero but I can't figure how to show it mathematically.

REPLY [7 votes]: $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2}) \leq (\sqrt{2}-1)^{n-2}$, done.