Show that a subset of $C[0,1]$ is closed I have a question I my metric spaces course book I cannot solve:

Show that the following subset is closed in $C[0,1]$: $\{ f \in C[0,1] \mid f(a)=0 \textrm{ for all }a \in A \}$, where $C[0,1]$ is the space of continuous real-valued functions on $[0,1]$ with the sup metric and $A \subseteq [0,1]$.
If $A$ was just one point, could you see why it's true? Can you see how this proves it for general $A$?
First proof it taking $A$ as a singleton which is shown by someone else above then it is obvious because intersection of closed sets is closed.