Expectation of exponential random variables

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Can someone help me understand the parts I have highlighted in the proof? I have not understood how $|2c...|$ is upper bounded by $1$ and how one computes the expectation of an exponential random variable? This claim is available on page 29 of https://arxiv.org/pdf/1707.00833.pdf I believe the "background material" is all available on the same page just slightly above the claims.

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