An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$ 
Prove that for any positive integer $n$, 
$$n^{1/n}  < 1 + \sqrt{\frac{2}{n}}.$$

This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29. 
Hint:

 Use the binomial theorem.


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