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I have been working on some Metric Space problems for roughly 20hrs now and I cannot seem to grasp some of these concepts so I was hoping someone here could clear a few things up for me. My first problem is detailed below...

I have the following metric...

d(x,y) = d(x,y)/(1 + d(x,y)

Now I did a search on google and found a few examples on how to solve this and even one on these forums; however, nothing seems to really make sense to me. On this post https://www.physicsforums.com/showthread.php?t=527353", I do not understand how to finally cancel terms out. I did the following..

a = d(x,y)

b = d(x,z)

c = d(y,z)

Plugging into the triangle inequality we get...

a ≤ b + c

From here we can use our metrics and get...

a/(1+a)≤ b/(1+b) + c/(1+c)

Now in the post it talks about multiplying each side by (1+a)(1+b)(1+c) If I do this I arrive at...

a(1+a)(1+b)(1+c)/((1+a)(1+a)(1+b)(1+c)) ≤ b(1+a)(1+b)(1+c)/((1+b)(1+a)(1+b)(1+c) + c(1+a)(1+b)(1+c)/((1+c)(1+a)(1+b)(1+c)

From here it just becomes a mess to me and I feel like I am not making any progress. Can someone point out to me what I am doing wrong and where I should be going with this please?

My next question is simple...

For proving the inverse triangle inequality I used d(z,y) → d(y,z) and substituted used that to switch out each value of x and y so for instance...

d(x,y) ≤ d(x,z) + d(y,z) → d(x,z) ≤ d(x,y) + d(z,y)

I was wondering if this was a valid method?

Thank you all in advance for you help!!

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# Metric Space Explanation

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