(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X be a metric space with metric d. Show that the space X, where distance is measured by d' = d/(1+d), is also a metric space.

2. Relevant equations

Three requirements of a distance function for X to be a metric space:

1. d(x,y) = 0 <=> x=y

2. d(y,x) = d(x,y)

3. d(x,z) <= d(x,y) + d(y,z)

3. The attempt at a solution

It's demonstrating the triangle inequality (3. above) that has me stumped.

My starting point is to try and use (1) d(x,z) <= d(x,y) + d(y,z) to demonstrate (2) d'(x,z) <= d'(x,y) + d'(y,z).

I tried manipulating the RHS of (2) to get a common denominator and then tried to 'bash' it out, without any success. I then tried to manipulate the inequalities, but I can't get (2) whilst maintaining <= as opposed to simply <.

The problem wasn't set as a difficult one, so I presume there is something simple that I can't seem to see.

Any help would be greatly appreciated!

Cheers

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# Simple Metric Question

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