Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$.
I'm having real trouble proving this inequality. I'd greatly appreciate any help.
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Finite or infinite number of $p_n$'s? – Hetebrij Feb 10 '16 at 16:09
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Infinite number – nomadicmathematician Feb 10 '16 at 16:12
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1Hint: Prove by induction on $n\geqslant1$ (or even $n\geqslant0$) that $$\prod_{k=1}^n(1-p_k)\geqslant1-\sum_{k=1}^np_k.$$ – Did Feb 10 '16 at 16:17