$$a \in X \Rightarrow \exists Q \in C \ni a \in Q$$
or else C wouldn't cover X
since the Q's are open
$$a \in Q \Rightarrow \exists r>0 \ni B_r(a)\in Q$$
that should be pretty much it. I leave it to you to finish.
Note of course that if a is in the boundary of X then it will necessarily lie in a Q that is not entirely contained in X but that doesn't matter here.