Is U(x,y) = max { |x1-y1|, |x2-y2|} a Valid Metric?

[math-chick_41]

I need help on how to show something is a metric.
U: R2 x R2 ----> R where U(x,y) = max { |x1-y1|,|x2-y2|}
I can do all the properties except for the dreaded triangle inequality property. help me out please, thanks

 

[Galileo]

What are your ideas so far?
Here's the basic start, let a=|x1-y1|,b=|x_2-y2|,c=|x3-y3|. (a,b and c are three arbitrary non-negative numbers).

To show:
max(a,b)<=max(a,c)+max(b,c)

Looks like, if it isn't obvious, you can always exhaust different cases.