Exercise 2.17.a

Exercise 2.17.a: Consider the metric space $$\mathbb{R}$$ (with the usual Euclidean metric). Show that the point $$0$$ is a limit point of the set $$K = \{1/n : n\text{ is a positive integer }\}$$.

Jamie
Goal: Show that the point $$0$$ is a limit point of the set $$K=\left\{\frac{1}{n}:\textrm{n is a positive integer}\right\}$$.

Proof: Let S be some open set that contains $$0$$. Because S is open, there is an open ball $$B=B\left(0,\epsilon{}\right)$$ such that $$B\subseteq{}S$$ and $$\epsilon>0$$. By Exercise 1.14, the sequence $$\frac{1}{n}$$ converges to $$0$$ and therefore there is some point $$p\in{}K$$ that is also a member of $$B$$. Because $$B\subseteq{}S$$ and $$p\in{}B$$, $$p\in{}S$$. This proves that every open set that contains $$0$$ also contains an element of $$K$$. This point is non-zero because there is no positive integer $$n$$ that satisfies $$\frac{1}{n}=0$$. Therefore, 0 is a limit point of $$K$$.

--202.36.179.68 23:37, March 16, 2011 (UTC)