WebSep 5, 2024 · Theorem 4.6.3. Every compact set A ⊆ (S, ρ) is bounded. Proof. Note 1. We have actually proved more than was required, namely, that no matter how small ε > 0 is, … WebA set A R is bounded if there exists M>0 such that jaj Mfor all a2A. Theorem 3.3.4. A set K R is compact if and only if it is closed and bounded. Proof. Let Kbe compact. To show that Kis bounded, suppose that Kis unbounded. Then for every n2N there is x n2Ksuch that jx nj>n. Since Kis compact, the sequence (x n) has a convergent, hence bounded ...
Compact Sets in Metric Spaces - UC Davis
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