How can I prove the statement |a-b| < |a| + |b| using the triangle inequality?

[Seda]

Homework Statement

a and b are real numbers.

Show l a-b l < l a l + l b l

Homework Equations

Well, I know la+bl < lal + lbl by the triangle inequality.

The Attempt at a Solution

If I can prove that la-bl < la+bl, then I'm done, but that most recent inequality almost seems too intuitive to write a formal proof. I can use the definition of absolute value to create cases perhaps, but I always get lost and seem to go nowhere

Just to write the obvious stuff down.

la-bl = a-b if a-b> 0 (or a>b)
la-bl = b-a if a-b< 0 (or b>a)

la+bl = a+b if a+b>0 (or a>-b) .
la+bl = -a-b if a+b<0 ( or a<-b)

I can't seem how to go somewhere with this.

 

[Dick]

|a-b|=|a+(-b)|. Now you can use the triangle inequality.

 

[EnumaElish]

By triangle ineq., distance between a and b < distance between a and 0 + distance between 0 and b.