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# Metric Space Proofs

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Problem 1: Given the metric space (X, p), prove that

a) |p(x, z) - p(y, u)| < p(x, y) + p(z, u) (x, y, z, u is an element of X);

b) |p(x, y) - p(y, z)| < p(x, y) (x, y, z is an element of X).

These problems are from Metric Space. Please give formal proofs for both (a) and (b) based on the reference provided. Thank you.

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https://brainmass.com/math/triangles/metric-space-proofs-232641

#### Solution Summary

This solution provides a step by step response which illustrates how to approach solving metric space proofs. The solution is provided in an attached Word document. Two Word documents, with the exact same solution, have been provided, one in the 2003 format and the other in the 2007.

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## Real Analysis Textbook Metric Spaces

Please solve the problems in B.pdf file by using the textbook.

The topics covered in this problem set are:

1. Metric spaces. Basic concepts ( Sections 5.1-5.2 )

Problems: - Section 5--# 1,2,4,5,6,7
2. Convergence. Open and closed sets (Sections 6.1-6.6)

Problems: - Section 6--# 1,2,3,4,5,9,10
3. Complete metric spaces (Sections 7.1-7.4)

Problems: - Section 7--# 1,3,4,5,7,9
4. Contraction mappings (Sections 8.1-8.3)

Problems: - Section 8 --# 1,2,4,6
5. Topological spaces

a. Basic Concepts (Sections 9.1,9.3,9.5.9.6)

Problems: - Section 9 --# 1,2,5

b. Compactness (Sections 10.1-10.3)

c. Compactness in Metric spaces (Sections 11.1-11.4)

Problems: - Section 11 --# 1,2,3,4
d. Real functions on Metric and Topological Spaces(Sections 12.1.12.2)

Problems: - Section 12 --# 1,2,3,5,6

6. Linear spaces

a. Basic Concepts (Sections 13..1-13.6)

Problems: - Section 13 --# 1,2,3,5,6

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