Let $x_n = \frac{\sqrt{n^2+2}}{2n},n \in \mathbb{N^*} \ldots$
Prove that $|{x_n - \frac{1}{2}}| < \frac{1}{2n^2}$
Indication: Use the relationship: $\sqrt{1+\gamma} < 1 + \frac{\gamma}{2}$, $ \forall \gamma \in \mathbb{R^{*}_{+}}$
I'd appreciate it very much if any answers could indicate a heuristic or general mindset one should have when proving such a proposition.