Question: How to prove this inequality

Question

I have to prove this inequality.

$$\sqrt[n]{n} - 1 \lt \sqrt{\dfrac 2n} ~~\mbox{for}~~ n \gt 2$$

So far I reached this result:

$$n^{\dfrac {n+2}{n}} - n^{\dfrac {n+1}{n}} + n - n^{\dfrac {n+1}{n}} \lt 2$$

Answer 1

In the (using the binomial theorem) expansion of

$$\left(1 + \sqrt{\frac{2}{n}} \right)^n$$

there is $1$, and there is $${n \choose 2} \left( \sqrt{\frac{2}{n}}\right)^2 = n-1$$

and other $>0$ terms. So $\left(1 + \sqrt{\frac{2}{n}} \right)^n > n-1 + 1 = n$.