How do I prove |cosa - cosb| <= |a-b|?

[FlorenceC]

Homework Statement

I have no idea how to approach this question.

Homework Equations

The Attempt at a Solution

I suppose ∫ |cosa - cosb| < = |a-b|
sinb-sina <= b^2/2 - a^2/2
but now what do I do?

 

[FlorenceC]

nvm. I figured it out, it's a subtle trick with MVT

 

[gopher_p]

Since it looks like you've found a solution ... assuming $a\leq b$ gives $$\left|\int_a^b\sin x\ \mathrm{d}x\right|\leq\int_a^b|\sin x|\ \mathrm{d}x\leq\int_a^b1\ \mathrm{d}x$$.
If $a>b$, you just need to flip the limits on the last two integrals. The desired inequality isn't too incredibly difficult to get from there.

I like the MVT proof better, though. It's, like, one step.