I want to prove, using mathematical induction, the following proposition:
$$1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}\:\ge \sqrt{n}, \forall n \geq 1 \in \mathbb{N}$$
My thesis is:
$$\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}}\ge \sqrt{n+1}, \forall n \geq 1 \in \mathbb{N}$$
I've proved the inequality for $n=1$, but after that I'm not being able to do the rest :/
Thank you for the help!