The following is a problem from my analysis homework:
Let $A$ be an infinite set in $\mathbb{R}$ with a single accumulation point in $A$. Must $A$ be compact?
What I'm having trouble understanding is the hypothesis. How can an infinite subset of $\mathbb{R}$ contain only $1$ of its accumulation points? Take $[0,1)$ for example, its accumulation points are $\{x ~| ~x \in [0,1]\}$ correct? That means it contains an infinite number of its accumulation points (all but $1$ in fact).
For an example of such an $A$ that is compact, consider $A = \{0\}\cup\{1/n\colon n\in\mathbb{N}\}$.
For an example of such an $A$ that isn't compact, consider $A = \Bbb Z \cup\{1/n\colon n\in\mathbb{N}\}$.