Theorem 6.15 (Heine-Borel Theorem). Let A be a subset of Rstd. Then A is compact if and only if A is closed and bounded.
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Could you explain how to show 6.15 in detail?
![The next theorem completely characterizes the sets in \(\mathbb{R}_{\text{std}}\) that are compact. This theorem, known as the Heine-Borel Theorem, is one of the fundamental theorems about the topology of the line.
Recall a set \(A\) in \(\mathbb{R}^1\) is **bounded** if and only if there is a number \(M\) such that \(A \subset [-M, M]\).
**Theorem 6.15 (Heine-Borel Theorem).** Let \(A\) be a subset of \(\mathbb{R}_{\text{std}}\). Then \(A\) is compact if and only if \(A\) is closed and bounded.
**Theorem 6.17.** Every compact subset \(C\) of \(\mathbb{R}\) contains a maximum in the set \(C\), i.e., there is an \(m \in C\) such that for any \(x \in C\), \(x \leq m\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56be79ad-be6a-45f1-b897-58d23fd7e62d%2F3f71b0e3-a513-4627-bb5d-bc985e49b53d%2Frue647j_processed.png&w=3840&q=75)
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