(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm using the following definition of an accumulation point:

A point [itex]a \in \textbb{R}[/itex] is an accumulation point of a set [itex]A\subset \textbb{R}[/itex] if every [itex]\epsilon-[/itex]neighborhood of [itex]a[/itex] contains at least one element of [itex]A[/itex] distinct from [itex]a[/itex].

Now, given the set [itex]A=\{\frac{1}{n}:n\in \textbb{Z}^+\}[/itex], I'm trying to prove that the only accumulation point of [itex]A[/itex] is 0.

2. The attempt at a solution

I was able to prove that 0 is an accumulation point, but my question is about proving there can't be any others.

Intuitively, it makes sense to me because any element between 0 and 1 is either an element of [itex]A[/itex] or between two elements of [itex]A[/itex]. In either case, you can take [itex]\epsilon[/itex] small enough to not include any elements of [itex]A[/itex].

The problem I have is that I'm not sure how to formulate this in a way that would be considered rigorous.

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# Homework Help: How do you prove that some point is the *only* accumulation point in a set?

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