My problem is the following:

Find $\displaystyle \overline{A} \textrm{ for } A=\{(x,\sin(1/x))|x>0.\}$

I know that $\displaystyle \overline{A}=A\cup\{(0,y)|-1\leq y \leq 1,\}$ but I need to figure out & understand the ``how'' behind that answer, in other words, give a proof that A-closure is that set above. Thanks.