Show that the sequence $x_n=\sum\limits_{k=1}^n\frac1{\sqrt{k+1}+\sqrt{k}}$ is unbounded. Consider the sequence $\{x_n\}_{n\ge1}$ defined by $$x_n=\sum_{k=1}^n\frac{1}{\sqrt{k+1}+\sqrt{k}}, \forall n\in\mathbb{N}.$$ Is $\{x_n\}_{n\ge 1}$ bounded or unbounded. 
I solved the problem as stated in the answer posted by me. Is it possible to solve the problem in a more better and rigorous manner?

$\sqrt {k+1}+\sqrt k \leq \sqrt {2k}+\sqrt k<3\sqrt k$. Hence the given sum is at least $\sum\limits_{k=1}^{n} \frac 1 {3\sqrt k}$ Now use the fact that the series $\sum\limits_{k=1}^{n} \frac 1 {3\sqrt k}$ is divergent.
