Let $\displaystyle m>0$ be given. If for all $\displaystyle 0<s<m, D\bigcap B'_{s}(c) \not= \emptyset$, then for all $\displaystyle r>0$, $\displaystyle D\bigcap B'_{r}(c) \not= \emptyset$

I understand the definition of limit point and neighborhoods but I'm not sure how to use them to prove this. Any help would be appreciated.