Verifying the metric space e = d / (1 + d)

[pdonovan]

I'm trying to verify that if (M,d) is a metric space, then (N,e) is a metric space where e(a,b) = d(a,b) / (1 + d(a,b)). Everything was easy to verify except the triangle inequality. All I need is to show that:

a <= b + c
implies
a / (1 + a) <= (b / (1 + b)) + (c / (1 + c)

Any help would be greatly appreciated, thank you!

 

[tiny-tim]

hi pdonovan!

try writing everything as a fraction over (1 + a)(1 + b)(1 + c), and seeing what cancels

 

[pdonovan]

Thank you for the tip, but I still haven't figured it out : /

I have
a<=b+c
-->
a/[(1+a)(1+b)(1+c)] <= (b+c)/[(1+a)(1+b)(1+c)]
-->
a/[(1+a)(1+b)(1+c)] <= b/[(1+a)(1+b)(1+c)] + c/[(1+a)(1+b)(1+c)]

 

[tiny-tim]

(just got up :zzz: …)

no, start at the answer, and work backwards!

 

[AdrianZ]

like tiny-tim said, write down the triangle inequality for the new metric and assume it's true. then multiply both sides by (1+a)(1+b)(1+c) (since it's positive) and then "work backwards" as tiny-tim said. then see if you can reversely do all the steps from the opposite direction. if you can, (and you can), then you're done.