I’ve been watching lectures on the Epsilon Delta definition of limits for a while now. I’m struggling to understand what it means.

1st question: What does this statement mean: 0<|x−a|<δ0<|x-a|< \delta? Please explain this in simple terms.
2nd question: Why do we have to guess delta?
3rd question: How do we structure proofs for these sorts of statements?
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1 Answer
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1st question: Let's first look at the simpler inequality |x−a|<δ|x - a | < \delta. It means "xx is within distance δ\delta from aa". To make it easier, rewrite it in the two-sided form:
−δ