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

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.

Answer 1

$$(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2}) \leq (\sqrt{2}-1)^{n-2} $$ and clearly $$\lim_{n \to \infty } (\sqrt{2}-1)^{n-2} =0$$

since $0<(\sqrt{2}-1)<1$

so by the comparison test we win.

Answer 2

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