TITLE: Prove that the set consisting of isolated points is finite.
QUESTION [6 upvotes]: 1. Let A be a compact subset in $R^n$. Investigate whether the following assertion is
true or not: If A consists of isolated points only then A is finite. 
I couldn't demonstrate my answer.We know that when A is a compact subset then it is closed and bounded. I was looking and I found that it resembles this question Is a closed subset of isolated points in a compact set necessarily finite? I am still not fully convinced.

REPLY [3 votes]: If the points of $A$ are isolated, then each singleton $\{a\}$ is an open set in $A$, for all $a\in A$. Therefore, the collection of these singletons form an open covering of $A$. By compactness of $A$, there exists $a_1,\ldots, a_n \in A$ such that $A = \{a_1\} \cup \cdots \cup \{a_n\}$. Thus $A = \{a_1,\ldots, a_n\}$.