It seems my biggest nightmare has come to haunt me again! I first came across the formal proof of a limit $\epsilon-\delta$ in Calculus 1, I never truly mastered it since at the time it was just racking my brain. I have now begun self-teaching a course in sequences & series and it has already come up. It has also been emphasised to me that being able to understand and do this is fundamental for success in this course.
I'm almost 100% certain that I'd have no trouble applying the methods, I've seen in tutorials to prove a sequence converges. However, I don't understand how the $\epsilon-N$ proof actually proves a sequence converges; if that makes much sense.
Here are a few questions, I'd like to ask:
- What does this statement mean: $|a_n - L| < \epsilon $ L refers to the limit.
Why does this proof actually prove a sequence converges?
Why does $\epsilon>0$?
Those are the only questions that currently come to mind, also if anyone can provide any additional advice on how I can wrap my head around these proofs please feel free to post!