Suppose $\displaystyle lim_{x\longrightarrow x_{0}}f(x)=b$ and there is a $\displaystyle r>0$ such that $\displaystyle f(B'_{r}(a)) \subseteq [0,\infty)$. Show that $\displaystyle b\geq 0$.

I understand the definition of limit but I'm having trouble getting this proof started. Any help would be appreciated.