# Is it true that $\sin x \leq |x|$? How can I find more of these inequalities?

I'm trying to find useful inequalities I can make use of when applying the squeeze theorem for multivariable limits. Is there a name for them or some good source where I can find a bunch of them? I haven't had luck googling so far.

Also, I seem to remember this one: $$\sin x \le |x|$$ Is that one true? I've been searching online but wasn't able to find it.

For $|x| \geq 1$ it is clear. Else you can use the function $f(x)=x-\sin(x)$. The derivative is $1-\cos(x)$ which is non-negative, so for $x\in[0,1]$ $f\geq f(0)=0$. And on $[-1,0]$ you use the same idea with $g(x)=-x-\sin(x)$.

Consider the Taylor expansion:

$$\sin x = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}x^{2k+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

which is an alternating series with $x$ as an upper bound.