An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$

Question

Prove that for any positive integer $n$,

$$n^{1/n} < 1 + \sqrt{\frac{2}{n}}.$$

This is due to Viktors Linis, Crux Mathematicorum (which was at the time called Eureka), Vol. 2, No. 2, February 1976, p. 29.

Hint:

Use the binomial theorem.

Answer 1

$$\left(1+\sqrt{\frac{2}{n}}\right)^n>1+C_n^2*\frac{2}{n}=n$$