Heine-Borel Theorem for R proof
Question
Heine-Borel Theorem for R proof
Solution
Sure, I can explain the proof of the Heine-Borel Theorem.
Step 1: Let's start by stating the theorem. The Heine-Borel Theorem states that a subset of Euclidean space R^n is compact if and only if it is closed and bounded.
Step 2: First, we will prove the "if" part, which states that if a subset is compact, then it must be closed and bounded.
Step 3: To prove the boundedness, assume that the subset is not bounded. This means that for any positive integer k, there exists a point x in the subset such that the distance between x and the origin is greater than k.
Step 4: Now, consider a sequence of points {x_n} in the subset. Since the subset is not bounded, we can choose a subsequence {x_n_k} such that the distance between x_n_k and the origin is greater than k for each k.
Step 5: Since the sequence {x_n_k} is a subsequence of {x_n}, it must also converge to a limit point x. However, since the distance between x_n_k and the origin is greater than k for each k, the distance between x and the origin must be infinite, which contradicts the fact that x is in the subset.
Step 6: Therefore, we have shown that the subset must be bounded.
Step 7: Next, we will prove the closedness of the subset. Assume that the subset is not closed, which means that there exists a limit point x that is not in the subset.
Step 8: Consider a sequence of points {x_n} in the subset that converges to x. Since x is a limit point, there exists a subsequence {x_n_k} that converges to x.
Step 9: However, since the subset is not closed, x must not be in the subset. This contradicts the fact that {x_n_k} is a sequence in the subset that converges to x.
Step 10: Therefore, we have shown that the subset must be closed.
Step 11: Now, let's prove the "only if" part, which states that if a subset is closed and bounded, then it must be compact.
Step 12: Assume that the subset is closed and bounded. We need to show that every sequence in the subset has a convergent subsequence that converges to a point in the subset.
Step 13: Since the subset is bounded, every sequence in the subset is also bounded. By the Bolzano-Weierstrass Theorem, every bounded sequence has a convergent subsequence.
Step 14: Let {x_n} be a sequence in the subset. By the Bolzano-Weierstrass Theorem, there exists a subsequence {x_n_k} that converges to a point x.
Step 15: Since the subset is closed, x must be in the subset. Therefore, we have shown that every sequence in the subset has a convergent subsequence that converges to a point in the subset.
Step 16: Hence, we have proved both the "if" and "only if" parts of the Heine-Borel Theorem, which completes the proof.
Similar Questions
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