I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt:
Prove that a closed subset of a compact set is compact.
My Proof:
Let X be a compact set and let Y be a closed subset of X. Since X is compact, every $x\in X$ is contained in some ball $B$ in $\mathbb{R}^n$. Since $Y\subseteq X$, $y\in Y$ implies $y\in X$, so all elements of $Y$ are also contained in $B$. Thus, $Y$ is closed (by assumption) and bounded (since it is contained in a ball), so Y is compact.
Proof looks good! Perhaps you define compact as "closed and bounded", but usually people define it a different way (every open cover has a finite subcover) and the Heine-Borel Theorem gives you compact iff closed and bounded. In that case, it is worth saying, "so $Y$ is compact by the Heine-Borel theorem".