# What is an infinite subset of a compact set?

I am attempting to work on the following proof:

If $$E$$ is an infinite subset of a compact set $$K$$, then $$E$$ has a limit point in $$K$$.

I know that that this proof has been answered here already, but I am more interested in understanding the statement itself. I am just struggling to comprehend what $$E$$ is in this question... an infinite subset of a compact set. For some reason, I just cannot visualize this. If anyone out there is able to help me see through the fog, or even provide a specific example for me to think about, I would greatly appreciate it.

There are many possibilities. Take $$(0,1)\subseteq [0,1]$$ or the circle embedded as the equator of the sphere: $$S^1\subseteq S^2$$.
If you are familiar with it, take the middle thirds cantor set $$C\subseteq [0,1]$$.
• Thank you for the examples. Would it also be correct to say that $[0,1] \subseteq [0,1]$ works too? I know this seems trivial, but I guess I am just wondering if the infinite subset can be closed as well. – user554608 Dec 8 '18 at 5:38
There are many examples as mentioned in the other answer. By Heine-Borel Theorem compactness is equivalent to the condition of being closed and bounded. Take the unit interval $$I$$ for example, you can pick the sequence $$1,\frac{1}{2},...,1/n...$$ as an infinite subset, or any open(or closed) interval contained in $$I$$.