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**DontKnoMaff** For the exponential function $\displaystyle \exp(z) = \sum^{\infty}_{n=0} \frac{z^n}{n!}$, we know that $\displaystyle \exp(z+w)=\exp(z).\exp(w)$, and that, if $\displaystyle x \geq 0$, then $\displaystyle \exp(x) \geq 1+x$. Use these facts to prove that, if $\displaystyle 1 \leq n \leq m$, then

$\displaystyle \left|\prod^n_{j=1} (1+a_j) - \prod^m_{j=1} (1+a_j)\right| \leq \exp\left(\sum^n_{j=1} |a_j|\right) . \left(\exp\left(\sum^m_{j=n+1} |a_j|\right)-1\right)$.