Trignometric inequalities. I was working through a question on limits where the author suggested the inequality: 
$ |\sin(x)| \leq x $ in some deleted neighbourhood around 0,
I'm kind of unsure if the inequality is correct because in a neighbourhood around 0 we would have x<0, but $|\sin(x)| > 0 $ for all x in any neighbourhood around 0. Am i missing anything?
Edit:
1.) removed the line "which leads to:
$-x \leq \sin x \leq x $  "

You're right, when $x<0$ the inequality $|\sin x| \le x$ is obviously false, and likewise for $-x \le \sin x \le x$.
(But $|\sin x| \le |x|$ is true for all real $x$.)
