Prove that $\sqrt{n(n + 1)} \leq n + \frac{1}{2}$ for all $n \in \mathbb{N}$

Question

Is it valid to say that since $\sqrt{n(n + 1)} > 1$ and $n + \frac{1}{2} > 1$ that we can square both sides without having to assume anything about their inequality?

Also can someone show me how with induction or strong induction?

Answer 1

For all $n\in\mathbb{N}$, $$\sqrt{n(n+1)}<\sqrt{n^2+n+\frac{1}{4}}=\sqrt{\left(n+\frac{1}{2}\right)^2}=n+\frac{1}{2}$$

Answer 2

Squaring both sides you get $$ n^2+n\le n^2+n+\frac {1}{4}$$ which is true for all n.