Formal demonstration of $\left|\thinspace\sin{\frac{1}{x}}\right|\leq 1;\thinspace x≠0$

Question

Is there any mathematical demonstration of the fact: $$\left|\thinspace\sin{\frac{1}{x}}\right|\leq 1;\thinspace x≠0$$

Intuitively, I can justify the expression, but I was hoping if I could fetch a more formal definition.

Answer 1

HINT.- For all $x\ne0$ one has $1/x=y\in\mathbb R$ and it is well known that $|\sin(y)|\le1$