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Let $X_{1},X_{2},\ldots$ be i.i.d. random variables and let $S_{n}=X_{1}+ \cdots +X_{n}$. Given that 1 and $0<\sigma<\lambda$, how do I show that if \sup_{1\leq b \leq a'-a}P(|S_{b}|\geq \sigma)\leq \frac{1}{2}, then P( \sup_{a\leq b\leq a'} S_{b}\geq \lambda)\leq 2P(S_{a'}\geq \lambda - \sigma)?

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    Just what Dimitri said. Strongly suggested reading: [How-to-ask-a-homework-question](http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question).2011-11-20

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