If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are sequences converging to $x$ and $y$ respectively, show that the sequence $\left(x_{n}+y_{n}\right)$ converges to $x+y$.

If $x_{n} \neq 0$ for all $n$ and $x \neq 0$, show that the sequence $\left(\frac{1}{x_{n}}\right)$ converges to $\frac{1}{x}$.

(a) Find $\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+n}-n\right)$.

(b) Determine whether $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}}$ converges.

Justify your answers.