Analysis: Prove |x|+|y| is less than or equal to |x+y|+|x-y|

[Chinnu]

Homework Statement

Using the triangle inequality, establish that:

|x| + |y| \leq |x+y| + |x-y|

Homework Equations

|x + y| \leq |x| + |y|

The Attempt at a Solution

I have tried a few things, here are those that seem like they would be most useful:

|x + y| \leq |x| + |y|

\leq |x+y-y| + |y-x+x|

\leq |x+y| + |-y| + |y-x| + |x|

\leq |x+y| + |y| + |y-x| + |x| ...Note that |y-x| = |x-y|

Also,

|x-y| \leq |x| + |-y| = |x| + |y|

which might be able to be used in the middle inequality above.

I'm not sure what to do from here (or if I'm on the right track)

 

[I like Serena]

Hi Chinnu!

Try substituting x=u+v and y=u-v.
Note that you can find a u and a v for any x and y.