Using the observation that the absolute value of sin(x) is less than or equal to the absolute value of x, show that sin(x) is continuous in x = 0.
You probably want an $\displaystyle \epsilon-\delta$ proof.
$\displaystyle \forall \epsilon>0, \exists \delta>0, \forall x \in \mathbb{R}: |x|<\delta \Rightarrow |\sin(x)|<\epsilon$
Can you proof the above statement (with the given hint)?