Proving Compactness Without the Heine-Borel Theorem (Proof Verification)

Question

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:

Let $K \subset R^1$ consist of $0$ and the numbers $\frac{1}{n}$, for $n = 1,2,3,\dots$. Prove that $K$ is compact directly from the definition (without using the Heine-Borel theorem).

My Proof:

Define a sequence $(k_n)$ where $k_0 = 0$ and $k_n = \frac{1}{n}$, so that $\cup(k_n) = K$. Suppose $\{G_\alpha\}$ is an open cover of $K$. Then $\cup_{n=0}^{\infty}B_r(k_n)\subset \cup\{G_\alpha\}$ for infinitely many open balls of radius $r$. But, for any $r$, $B_r(k_0) = B_r(0)$ will contain infinitely many elements of $(k_n)$ as shown:

$(k_n) = \frac{1}{n} < 0 + r = r \to \frac{1}{r} < n$

So, all values of $(k_n)$ where $n > \frac{1}{r}$ are contained in $B_r(0)$. Thus, $\cup_{n=0}^{\frac{1}{r}}B_r(k_n)$ covers $\cup(k_n)$ and $\cup_{n=0}^{\frac{1}{r}}B_r(k_n)\subset \cup\{G_\alpha\}$. Therefore, it is a finite subcover of $K$ and so $K$ is compact.

Answer 1

Elements that are lacking clarity in your proof:

• You haven’t define what the $B_r(k_n)$ are.
• Reading between the lines, we can suspect that those are open balls with radius equal to $r$ centered on $k_n$.
• Which leads to following issues: (1) an open cover maybe by something else than balls, (2) the balls may have different radius and (3) may not be centered on the $k_n$.

For a proof, I would say that if an open cover $\mathcal U$ covers $K$, then it will exists an open $U \in \mathcal U$ such that $0 \in U$. Then prove that $U$ contains all but a finite number of elements of $K$. Conclude from there.