Inequality in proof of 2nd Borel-Cantelli Lemma

Question

At some point in the proof of the second Borel-Cantelli Lemma the the following inequality is mentioned:

$$...=\exp\bigl ( \sum_{m=n}^k\log(1-P(A_m)\bigr ) \leq \exp \bigl (-\sum_{m=n}^kP(A_m) \bigr)$$

How do I this inequality? Some simple logarithmic calculation rules?

Answer 1

Note that

• $\log (1-x) =- \sum_{n=1}^{\infty}\frac{x^n}{n} \leq -x$ for $0 \leq x<1$ $$\Rightarrow \exp\bigl ( \sum_{m=n}^k\log(1-P(A_m)\bigr ) \leq \exp\bigl ( \sum_{m=n}^k -P(A_m)\bigr ) = \exp \bigl (-\sum_{m=n}^kP(A_m) \bigr)$$