Chapter 7: Completeness and Compactness

Question 1

Cauchy Sequence

Answer 1

Let (X,d) be a metric space. A sequence {xn} in X is a Cauchy seuence if for every epsilon greater than 0 there exists an M in Naturals such that for all n greater than or equal to M and all k greater than or equal to M we have d(xn, xk) < epsilon

Question 2

Facts about Cauchy Sequences in Metric Spaces

Answer 2

1. A convergent sequence in a metric space is Cauchy

2. The space Rn with the standard metric is complete

Question 3

Complete, or Cauchy Complete

Answer 3

Let (X,d) be a metric space. We say X is complete or Cauchy complete if every cauchy sequence {xn} in X converges to an x in X.

Question 4

Compact

Answer 4

Let (X,d) be a metric space and K in X. The set K is said to be compact if every open cover of K has a finite subcover.

Question 5

Facts about Compactness

Answer 5

1. Let (X,d) be a metric space. A compact set K in X is closed and bounded
2. Let (X,d) be a metric space. Then K in X is a compact set if and only if every sequence in K has a subsequence converging to a point in K.
3. Let (X,d) be a metric space and let K in X be compact. If E in K is a closed set, then E is compact.

Question 6

Lebesgue Covering Lemma

Answer 6

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Question 7

Heine Borel Theorem

Answer 7

A closed, bounded subset K in Rn is compact