One of the proofs given in my course on metric spaces is the following:
Claim
Consider the Hilbert space $ℓ^2$ of square-summable real sequences with inner product
$$\langle x,y\rangle = \sum^{\infty}_{n=1} x_n y_n $$ For $x = (x_n)_{n \in \mathbb N}$ and $y = (y_n)_{n \in \mathbb N}$.
Then the set $B = \{x \in ℓ^2 : \|x\|=1\}$ is closed.
Proof
The function $\| \bullet \|$ is continuous, therefore the set $B$ is closed.
My question is, why does the continuity of the norm directly imply that $B$ is closed, and does this hold in general, for all continuous functions and metric spaces?