Spivak Sine confusion (possible error) quote from Spivak:
"Let us consider the function $f(x) = \sin(1/x)$." The goal is to show it is false that as $x \to 0$ that $f(x)\to 0$
He says we have to show "we simple have to find one $a > 0$ for, which the condition $f(x) < a$ cannot be guaranteed, no matter how small we require $|x|$ to be. In fact $a = 1/2$ will do. It is impossible to ensure that $|f(x)| < 1/2$ no matter how small we require $x$ to be." (Spivak 93). 
Okay, major issue here. 
$\sin(1/x) < 1/2$
$ \implies 1/x < \arcsin(1/2)$
$ \implies 1/x < \pi/6$
$ \implies 6/\pi < x$
I just proved $\sin(1/x)$ can be less than 1/2. what is Spivak talking about?!?

Your first step is not correct: arcsin just gives you one possible argument, not the only one.
Example: you can have $sin(x)<0$ but not $x<arcsin(0)=0$, just take $x=3\pi/2$.
