prove that a compact set is closed using the Heine - Borel theorem

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Colby Messinger answered 2 years ago

Prove the Heine-Borel Theorem

Prove that a subspace of R is compact if and only if it is closed and bounded.

1. Prove the Heine-Borel Theorem (Theorem 3.35). 2. Suppose f: X → Y maps from the...

1. Prove the Heine-Borel Theorem (Theorem 3.35). 2. Suppose f: X → Y maps from the metric space X to the metric space Y, and x ∈ X. Prove that f is continuous at x if and only if, for any sequence {xn} in X that converges to x, f(xn) → f(x).

1. Prove that any compact subset ot a metric space is closed and bounded. 2. Prove...

1. Prove that any compact subset ot a metric space is closed and bounded. 2. Prove that a closed subset of a compact set is a compact set.

Prove that the intersection of two compact sets is compact, using criterion (2).

Question: Prove that the intersection of two compact sets is compact, using criterion (2). Prove that the intersection of two compact sets is compact, using criterion (1). Prove that the intersection of two compact sets is compact, using criterion (3). Probably the most important new idea you'll encounter in real analysis is the concept of compactness. It's the compactness of [a, b] that makes a continuous function reach its maximum and that makes the Riemann in- tegral exist. For...

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prove that a compact set is closed using the Heine - Borel theorem

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