I Chernoff Bounds for Independent Bernoulli Sums

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WMDhamnekar
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What is wrong with this proof? Can you notice that? or I am wrong. In my opinion, in the R.H.S. of inequality (3.2), the index of 'e' must be positive if we use the proof. I also want to know how to derive the proof of inequality(3.3)? Author said it is similar to that of (3.2). But I don't understand that.
Chernoffbounds proof.png
 
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I cleared my doubt taking suitable guidelines from other statistician on Internet.
 

Thread 'How does E[X] and E[|X|] relate?'

Greetings, I am studying probability theory [non-measure theory] from a textbook. I stumbled to the topic stating that Cauchy Distribution has no moments. It was not proved, and I tried working it via direct calculation of the improper integral of E[X^n] for the case n=1. Anyhow, I wanted to generalize this without success. I stumbled upon this thread here: https://www.physicsforums.com/threads/how-to-prove-the-cauchy-distribution-has-no-moments.992416/ I really enjoyed the proof...
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