Show that $\exp (x) \geqslant 1+x$ for $x \geqslant 0$.

Let $\left(a_{j}\right)$ be a sequence of positive real numbers. Show that for every $n$,

$$\sum_{1}^{n} a_{j} \leqslant \prod_{1}^{n}\left(1+a_{j}\right) \leqslant \exp \left(\sum_{1}^{n} a_{j}\right)$$

Deduce that $\prod_{1}^{n}\left(1+a_{j}\right)$ tends to a limit as $n \rightarrow \infty$ if and only if $\sum_{1}^{n} a_{j}$ does.