Heine-Borel Theorem; If $E \subset \mathbb{R}^k$, then $E$ is compact iff $E$ is closed and bounded.
I have proved 'closed and bounded⇒compact' and 'compact⇒bounded'.
(There exists $r\in \mathbb{R}$ such that for every $x\in E$, $|x| The proof in Rudin PMA p.40 uses 'countable axiom of choice' I have googled it and found some proofs, but they all used some weaker form of AC. Please help me how to show that $compact⇒closed$ in ZF..