Purchase Solution

Real Anaylsis : Metrics Proof

Not what you're looking for?

Ask Custom Question

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.

Purchase this Solution
Solution Summary

A metrics proof is provided. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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.
...

Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Probability Quiz

Some questions on probability

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

View More Free Quizzes
Related BrainMass Solutions

  • Bounded Real Sequences and Metrics

    34598 Bounded Real Sequences and Metrics Let X be the set of all bounded real sequences ... (SEE ATTACHMENT FOR COMPLETE QUESTION) ... For points x = {x1,,x2,x3...} and y = {y1,y2,y3...} in X define: d(x,y) := SUP |xk-yk|.

  • Bounded Sequences, Metrics and Completeness

    87765 Real Analysis : Bounded Sequences, Metrics and Completeness Let X be the set of all bounded sequences of real numbers.

  • Integrable functions

    10528 Real Anaylsis Let f: [a,b] be mapped onto the Reals be a function that is integrable over [a,b] and let g: [a,b] be mapped onto the Reals be a function that agrees with f except at finitely many points. Is g integrable over [a,b]?

  • How does software engineering differ?

    is that in mechanical and electrical engineering, engineers construct real artifacts such as bridges and circuits, while software engineers create virtual artifacts.

  • Real Analysis

    It is enough to show this in the interval [0,1] , as then we can extend the same proof to any interval [K,K+1]. In the interval [0,1] we represent real numbers by infinite decimal fractions, r=0.d_1 d_2...d_n.... where d_i are the digits.

  • Metrics and Euclidean n-Space

    Metrics and Euclidean n-Space are investigated. The solution is detailed and well presented.

  • Real Analysis : Real Numbers

    163176 Real Analysis : Real Numbers College level proof before real analysis. Please give formal proof. Please explain each step of your solution. If you have any suggestion or question to me, please let me know. Thank you.

  • Proof Techniques

    270490 Proof Techniques: Proof with Counterexample Proof Techniques: Homework 03 Provide counterexamples to each of the following. Every odd number is prime. Every prime number is odd. For every real number x, we have x2 > 0.

  • Real Analysis: Proof by induction that a polynomial of degree n>0 has at most n roots.

    Note: We only consider polynomial with REAL coefficients, and roots are only real roots, since every polynomial of degree n>0 with complex coefficients has exact n roots in complex regions. Proof.

View More Related Solutions