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.

Expert Solution

Colby Messinger answered 2 years ago

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

Let A be a subset of all Real Numbers. Prove that A is closed and bounded...

Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in...

Suppose K is a nonempty compact subset of a metric space X and x ∈ X....

Suppose K is a nonempty compact subset of a metric space X and x ∈ X. (i) Give an example of an x ∈ X for which there exists distinct points p, r ∈ K such that, for all q ∈ K, d(p, x) = d(r, x) ≤ d(q, x). (ii) Show, there is a point p ∈ K such that, for all other q ∈ K, d(p, x) ≤ d(q, x). [Suggestion: As a start, let S = {d(x,...

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

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