Prove Inequality: llxl - lyll < lx - yl

[Fairy111]

Homework Statement

For real numbers x and y prove the following:

llxl - lyll < (or equal to) lx - yl

Homework Equations

The Attempt at a Solution

Im not really sure where to start, i was considering cases where x < 0 and say y< 0 and what that would imply say x-y would be. But I am not sure how to continue.

 

[gabbagabbahey]

Hint: What is $(|x|-|y|)^2$?...How about $(|x-y|)^2$?

 

[icantadd]

Homework Statement

For real numbers x and y prove the following:

llxl - lyll < (or equal to) lx - yl
.

You may need to consider a couple cases. There may be a little trick to doing this one.
I would say look at | x | equals and then look at what | y | equals.
So maybe add an subtract some thing to x, do the same for y. Then the triangle inequality says that
| a + b | <= |a| + |b| . This should be helpful in solving this. Does that help?

 

[Fairy111]

ok, thankyou - i will have ago, althought I am not very good at proving things! Also what does the double modulus sign mean?

 

[icantadd]

ok, thankyou - i will have ago, althought I am not very good at proving things! Also what does the double modulus sign mean?

Me either, I think it just takes a lot of patience. I may be misunderstanding your question, but I believe the fact that there is this nested absolute value in the inequality is done to provide a relationship that you will find useful, especially with sequences. I remember a couple of proofs that I would not have gotten without knowing the statement you are trying to prove.

Conceptually though it makes sense. Think about the modulus (absolute value) sign as a measure. Then what the statement below is saying something obvious. That if you have take the absolute value of some number A it will always be positive, likewise for B. So | A | - | B | is guaranteed to be less than A. On the other hand A - B is not guaranteed to be less than A, suppose A is positive and B is negative. And so the distance between | |A| - |B| | <= | A - B|. Or at least, that is how I conceptualize absolute values. There is also an inequality way to interprer them. Let k be some positive number. Then | x | <= k iff
-k <= x <= k