MHB Intro to Logic (prove sequents)

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The discussion focuses on proving specific logical sequents, with participants expressing difficulty in starting the proofs. The first sequent involves demonstrating the relationship between existential and universal quantifiers. The second sequent requires proving the negation of an existential statement based on a universal condition. The third sequent connects the existence of a property with its implication to another property. Overall, the thread emphasizes the challenges of understanding and applying logical proofs in formal logic.
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I am stuck on these questions and don't really know how to start/solve them.
prove the following sequent:
1. $(\exists x) Fx \to (\forall x) Gx \vdash (\exists x)(Fx \to (\forall x)Gx)$

2. $(\forall x)(Fx \to (\forall y)\neg Fy) \vdash \neg(\exists x)Fx$

3. $(\exists x)Fx, (\forall x)(Fx \; à \; Gx) \vdash (\exists x)G$
 
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Thread 'How do 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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