Let $E$ be a bounded subset of $\mathbb{R}$. Prove that $\overline{E}$ is compact.
I am trying to prove this, but I keep getting stuck. I know that if $E$ is closed and bounded then I can conclude that $E$ is compact. I also know that $\overline{E}$ is closed, but I need to show $E$ is closed. However, I don't understand how I would do this. Could someone please demonstrate how to show $E$ is closed.
Note(Definition): If $E$ is a subset of $\mathbb{R}$, let $E'$ denote the set of limit points of $E$. The closure of $E$, denoted $\overline{E}$ is defined as $\overline{E}= E \cup E'$.
Best Answer
You do not need to show that $E$ is closed (as what @)carmichael561 said in his comment.)
Well, $E$ is assumed to be bounded and so, we can find a closed interval $I$ such that $E\subset I$. This implies that $I$ is a closed set that contains $E$. But $\overline{E}$ is the smallest closed set that contains $E$. Thus, $\overline{E}\subset I$. This implies that $\overline{E}$ is bounded. Since $\overline{E}$ is also closed, it follows from the Heine-Borel Theorem that $\overline{E}$ is compact.