- #1
Chinnu
- 24
- 0
Homework Statement
Show that:
(|x+y|)/(1+|x+y|) ≤ ((|x|)/(1+|x|)) + ((|y|)/(1+|y|))
Homework Equations
You are given the triangle inequality:
|x+y| ≤ |x| + |y|
The Attempt at a Solution
(This is done from the result, as I haven't been able to find the starting point)
(|x+y|)/(1+|x+y|) ≤ (|x|(1+|y|)+|y|(1+|x|))/((1+|x|)(1+|y|))
(|x+y|)/(1+|x+y|) ≤ (|x|+2|x||y|+|y|)/(1+|x|+|y|+|x||y|)
This doesn't seem to go anywhere. I also tried flipping the whole thing to get:
(1+|x+y|)/(|x+y|)≤(1+|x|)/(|x|)+(1+|y|)/(|y|)
but this doesn't seem to lead anywhere either...
I'm not sure how to go about this problem.