I found a question which has asked to prove or disprove any closed set of $\mathbb{R}$ is compact.
My Attempt
Since $\mathbb{R}$ is a set of itself and it is closed ($\mathbb{R}$ is open and closed too. So I though I can get it as closed one here) it is closed. That means any closed set of $\mathbb{R}$ is closed. But the problem is that Any set of $\mathbb{R}$ is not bounded since $\mathbb{R}$ itself is not bounded. So I think that statement is false.
Problem
But I don't know whether my proof is true or false. If someone can please help me to figure this out.
Note : Problem has clearly stated that "Any set of $\mathbb{R}$"
Best Answer
You are correct. The Heine-Borel Theorem states that a subset of $\mathbb{R}$ is compact if and only if it is closed and bounded. So, you a specific counter-example you just need to exhibit a subset of $\mathbb{R}$ which is closed and unbounded. For instance $[0,\infty)$.