Prove $(1 + \sqrt n)^{2/n} \leq (1 + 1/\sqrt n )^2$ for all natural $n$. Prove $(1 + \sqrt n)^{2/n} \leq (1 + 1/\sqrt n )^2$ for all natural $n$
I may or may not have to use Bernoulli's Inequality in this question.
I tried using Bernoulli's inequality on both sides of the question but I got nothing so far.
Am I even on the right path?

It's simpler than that.  Taking each side to the $n/2$th power, your inequality is equivalent to
$$1+\sqrt n \le \left(1+{1\over\sqrt n}\right)^n$$
But the right hand side expands as
$$\left(1+{1\over\sqrt n}\right)^n=1+{n\over\sqrt n}+\text{more terms, all positive}$$
