Is $x_n =\sqrt{n} (\sqrt{n+1}− \sqrt{n})$ , $n \ge 1$ convergent? $$x_n =\sqrt{n} (\sqrt{n+1}− \sqrt{n}) ,\quad n \ge 1$$
My work is:
First i analyzed the convergence of this sequence in 2 parts:
a) $\lim \sqrt{n+1}− \sqrt{n} = \lim \frac{(\sqrt{n+1}− \sqrt{n})(\sqrt{n+1}+\sqrt{n})}{\sqrt{n+1}+\sqrt{n}} = \lim \frac{n+1-n}{\sqrt{n+1}+\sqrt{n}} = \lim \frac{1}{ \infty} = 0$ when $n  \rightarrow \infty$.
But now how can I conclude that $\lim_{n  \rightarrow \infty}x_n = 0$

Simply
$$x_n=\sqrt n(\sqrt{n+1}-\sqrt n)=\frac{\sqrt n}{\sqrt{n+1}+\sqrt n}=\frac{\sqrt n}{\sqrt n\left(\sqrt{1+\frac{1}{n}}+1\right)}$$
now cancel and pass to the limit.
