Consider a sequence $\left(a_{n}\right)_{n \geqslant 1}$ of real numbers. What does it mean to say that $a_{n} \rightarrow$ $a \in \mathbb{R}$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$ ? What does it mean to say that $a_{n} \rightarrow-\infty$ as $n \rightarrow \infty$ ? Show that for every sequence of real numbers there exists a subsequence which converges to a value in $\mathbb{R} \cup\{\infty,-\infty\}$. [You may use the Bolzano-Weierstrass theorem provided it is clearly stated.]

Give an example of a bounded sequence $\left(a_{n}\right)_{n \geqslant 1}$ which is not convergent, but for which

$$a_{n+1}-a_{n} \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty$$