# Real Anaylsis : Metrics Proof

Let d be a metric in X. Prove that p(x,y)=(d(x,y))/(1+d(x,y)) is also a metric in X.

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#### Solution Preview

Proof:

From the condition, d is a metric in X. To show that p is a metric in X, we have to verify the followings.

(1) p(x,y)>=0.

Since d(x,y)>=0, then p(x,y)=d(x,y)/(1+d(x,y))>=0.

(2) p(x,y)=0 if and only if x=y.

If p(x,y)=d(x,y)/(1+d(x,y))=0, then d(x,y)=0, then x=y because d is a metric.

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#### Solution Summary

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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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