Mean Value Theorem and/or Rolle's Theorem

Question

Prove that the absolute value of cos(a)-cos(b) is less than or equal to the absolute value of a-b for all a and b

Answer 1

Mean Value Theorem:

(cos a - cos b)/(a -b) = -sin c for some value of c between a and b

|cos a - cos b| = |-sin c||a - b|

but since |-sin c| is less than of equal to 1,

|cos a - cos b| is less than or equal to |a - b|