# Limit points of $\left\{\frac{1}{n}+\frac{1}{m}:m,n\in{\Bbb N}\right\}$?

Let $$S=\left\{\frac{1}{n}+\frac{1}{m}:m,n\in{\Bbb N}\right\}$$ and $S'$ be the set of limit points of $S$. All the results I've found on Google or Math.SE only give the following $$\left\{\frac1n:n\geq 1\right\}\cup\{0\}\subset S'.$$

Here is my question: Is $$\left\{\frac1n:n\geq 1\right\}\cup\{0\}\supset S'$$ also true?

• Yes; here's a sketch: Convince yourself the only limit points must be in $[0,1]$. Now pick a point in $[0,1]$. If it's of the form $1/n$ or $0$, then we're done. Otherwise, the point must fall in $(\frac{1}{k+1}, \frac{1}{k})$ for some $k \in \mathbb{N}$. Finally, convince yourself we cannot get arbitrarily close to such a point. (Posted as a comment since many details are omitted.) – Benjamin Dickman Oct 4 '13 at 1:01
• But $S$ contains $2$? – Tunococ Oct 4 '13 at 1:12
• @Tunococ: according to wikipedia it is demanded that any neighbourhood of the limit point contain a point of the set other than the limit point if that happens to lie in the set (so points like 5/6 etc are also excluded). – doetoe Oct 4 '13 at 1:18
• @doetoe Oh you're right. That definition certainly makes more sense in the context of this problem. – Tunococ Oct 4 '13 at 8:35
• A COMPLETE PROOF IS HERE: math.stackexchange.com/questions/930646/… – Gregory Grant Apr 2 '15 at 14:01

Such a sequence $(a_i)$ is of the form $({1\over n_i} + {1\over m_i})$, so we have two sequences of natural numbers $(n_i)$ and $(m_i)$ (not uniquely determined by $(a_i)$ but that doesn't matter).
If one of these can get arbitrarily large, say a subsequence of $(n_i)$ goes to infinity, then a subsequence of $(a_i)$ goes to the limit of the corresponding subsequence of the sequence $({1\over m_i})$, which exists, because otherwise the original sequence wouldn't have a limit either. This gives the limit points of the form ${1\over m}$.
If neither grows arbitrarily large, there is only a finite number of possibilities for $a_i = {1\over n_i} + {1\over m_i}$, so the limit must be an element of the sequence, and by definition is not a limit point.