Heine-Borel Theorem
The theorem connecting the concept of compactness with boundedness and closedness.
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Theorem 1 (Theorem of Heine-Borel). For a subset \(C\subset\mathbb{K}\), the following statements are equivalent:
\(C\) is compact;
\(C\) is bounded and closed.
Proof:
“(i)\(\Rightarrow\)(ii)”: Let \(C\) be compact.
Let \((a_n)_{n\in\mathbb{N}}\) be a convergent sequence in \(\mathbb{K}\) with \(a_n\in C\) and \(a:=\lim_{n\rightarrow\infty} a_n \in\mathbb{K}\). Since \(C\) is compact, there is a subsequence \((a_{n_k})_{k\in\mathbb{N}}\) such that \(b:=\lim_{k\rightarrow\infty} a_{n_k}\in C\). By the theorem on the convergence of subsequences, every subsequence of a convergent sequence is also convergent having the same limit which gives us \(a=b\in C\). This shows that \(C\) is closed.
Now assume that \(C\) is unbounded.
Then for all \(n\in\mathbb{N}\), there
exists some \(a_n\in C\) with \(|a_n|\geq n\). Consider an arbitrary
subsequence \((a_{n_k})_{k\in\mathbb{N}}\). Due to \(|a_{n_k}|\geq n_k\geq k\), we have that
\((a_{n_k})_{k\in\mathbb{N}}\) is
unbounded, i.e., it cannot be convergent. This is also a contradiction
to compactness.
“(ii)\(\Rightarrow\)(i)”: Let \(C\) be closed and bounded. Let \((a_n)_{n\in\mathbb{N}}\) be a sequence in
\(C\). The boundedness of \(C\) then implies the boundedness of \((a_n)_{n\in\mathbb{N}}\). By the Theorem of
Bolzano-Weierstraß, there exists a convergent subsequence \((a_{n_k})_{k\in\mathbb{N}}\), i.e., \[\lim_{k\to\infty}a_{n_k}=a\] for some
\(a\in\mathbb{K}\). For compactness, we
now have to show that \(a\in C\).
However, this is guaranteed by the closedness of \(C\).\(\Box\)
Remark 2. Taking a closer look to the proof “(i)\(\Rightarrow\)(ii)’, we did not explicitly use that we are dealing with one of the spaces \(\mathbb{R}\) or \(\mathbb{C}\). Indeed, the implication that compact sets are bounded and closed holds true for all normed spaces. However, “(ii)\(\Rightarrow\)(i)” does not hold true in arbitrary normed spaces. Indeed, there are examples of normed spaces that have bounded and closed subsets which are not compact.
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