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I have a question about the proof of Proposition 3.14 from the book of Kallenberg (Foundations of Modern Probability). I will state the theorem first and explain which part of the proof I am not getting.

Let $\xi_1,\xi_2,\ldots$ be independent nonnegative random variables. Then $\sum_n \xi_n < \infty $ a.s. if and only if $\sum_n E[\xi_n \wedge 1] < \infty$.

Proof of $\implies$: $\sum_n \xi_n < \infty $ a.s. $\implies$ $\sum_n \xi_n \wedge 1 < \infty$. Hence wlog we can assume $\xi_n \leq 1$ for all $n$. By the inequalities $1-x \leq e^{-x} \leq 1 - (1-e^{-1})x$ for $x \in [0,1]$, we get $$0 < E[e^{-\sum_n \xi_n}] = \prod_n E[e^{-\xi_n}] \leq e^{-(1-e^{-1})\sum_nE[\xi_n]}$$

I understand all the steps so far but I don't see how we conclude $\sum_nE[\xi_n]$ from this.

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    If $\sum E[\xi_n]$ diverges, the upper bound on the right, which should read $$e^{-(1-e^{-1})\sum E[\xi_n]}$$ (you have a typo), is $0$, which is impossible since the lower bound $0$ on the left is strict. – 2017-01-03
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    @Did Thank you as always. – 2017-01-03

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