Use the Mean Value Theorem to prove the inequality|sin(a)−sin(b)| ≤ |a−b|for all real numbers a and b.

Answer 1

The mean value theorem states that there exists some c∈(a,b) such that

|sin(a) - sin(b)| = |cos(c)| |a - b|.

But since |cos(x)| ≤ 1, we have

|sin(a) - sin(b)| = |cos(c)| |a - b| ≤ 1 · |a - b| = |a - b|

thus |sin(a) - sin(b)| ≤ |a - b| as desired.