Big-O Estimate for Real Function/Examples/Sine Function at Infinity

Example of Big-$\OO$ Estimate for Real Function
Let $f: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map f x = \sin x$

Then:
 * $\map f x = \map \OO 1$

as $x \to \infty$.

Proof
Let us consider the real function $g: \R \to \R$ defined as:
 * $\forall x \in \R: \map g x = 1$

Then we have that:
 * $\forall x \in \R: \size {\map f x } < 2 \cdot \size 1$

Hence the result.