Proof involving Lipschitz continuity with constant L

Problem:

Prove that if the function f:[a,b]-->R is Lipschitz continuous with constant L then for every partition P of [a,b]

U(f,P)-L(f,P) <= ||P||*L*(b-a)

Okay so from Lipschitz continuity I see that dY(f(a),f(b)) <= L*dx(a,b) for all a,b in X

So dx(a,b) is the same as (b-a)

Then take a partition P with ||P|| < delta for both sides

Where do i go from here .... i nearly have the right side completed .... or so i think but what should my next move be when it comes to the left side? How does dY(f(a),f(b)) become U(f,P)-L(f,P)?

PLease help