So I have this inequality and I just can't figure out how to prove it:
Prove that ($\forall n\in \mathbb N)$ $$\sum_{k=1}^n \frac{1}{(k+1)\sqrt k}<2.$$ I've figured that for $n=1$ the inequality holds, since: $\frac{1}{2}<2$; so the statement is true for some $n \in \mathbb N$. Although I can't figure out how to prove the implication, that if the statement holds for some $n \in \mathbb N$, then it also holds for the number $n+1$.
I can't seem to make use of the induction hypothesis since the right-hand side is just a constant.
I'd be grateful for any suggestions.
edit: Solved, thanks for the answers.