Prove that this set is closed

Question

Let $C$ be the set of continuous real valued functions on $[0,1]$ with metric

$$d(f,g) = \text{sup} \{ |f(x) - g(x)| \ : x \in \ [0,1]\}$$

Show that $X = \{ f \ \in \ C \ : \ f(0) =0 \}$ is a closed subset of $C$.

I don't know how to prove this one.

Please help.
Thanks in advance.

Answer 1

For $(f_{n})\subseteq X$ such that $d(f_{n},f)\rightarrow 0$, then $|f(0)|\leq|f_{n}(0)-f(0)|+|f_{n}(0)|=|f_{n}(0)-f(0)|\leq d(f_{n},f)\rightarrow 0$, so $f(0)=0$.