Modulus of Sine of x Less Than or Equal To Absolute Value of x

Theorem
Let $x$ be a real number.

Then:
 * $\size {\sin x} \le \size x$

Proof
Clearly the inequality holds if $x = 0$.

Take $x \ne 0$.

From the Mean Value Theorem and Derivative of Sine Function, there exists $c \in \R$ such that:


 * $\ds \frac {\sin x - \sin 0} {x - 0} = \cos c$

so:


 * $\sin x = x \cos c$

Then we have:

as required.