proof of Simultaneous converging or diverging of product and sum theorem

From the fact that $1+x\le e^x$ for $x\ge 0$ we get

$$\sum _{n=1}^m{a}_n\le \prod _{n=1}^m(1+a_n)\le e^{{\sum }_{n=1}^m{a}_n}$$

Since $a_n\ge 0$ both the partial sums and the partial products are monotone increasing with the number of terms. This concludes the proof.