Suppose $a_{n} \in \mathbb{R}$ for $n \geqslant 1$ and $a \in \mathbb{R}$. What does it mean to say that $a_{n} \rightarrow a$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$ ?

Show that, if $a_{n} \neq 0$ for all $n$ and $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$, then $1 / a_{n} \rightarrow 0$ as $n \rightarrow \infty$. Is the converse true? Give a proof or a counter example.

Show that, if $a_{n} \neq 0$ for all $n$ and $a_{n} \rightarrow a$ with $a \neq 0$, then $1 / a_{n} \rightarrow 1 / a$ as $n \rightarrow \infty$.