Showing the continuity of some function on normed linear spaces

Question

Let $A$ be a subset of a normed linear space $V$. Define the distance from $\beta \in V$ to $A$ to be the real number
$$\rho(\beta,A) = \operatorname{glb} \{ \|\beta-\alpha\| : \alpha \in A \}.$$ How would you show that $\rho(\cdot,A) : V \to \mathbb{R}$ is continuous and that $\beta \in \bar{A}$ iff $\rho(\beta,A)=0$?

Answer 1

First part: If $||\beta_1 - \beta_2|| < \epsilon$, then for any point $\alpha \in A$, $||\beta_2 - \alpha|| \leq ||\beta_1 - \alpha|| + \epsilon$ by the triangle inequality. It follows that $\rho(\beta_1, A) < \rho(\beta_2, A) + \epsilon$. Symmetrically, $\rho(\beta_2, A) \leq \rho(\beta_1, A) + \epsilon$. It follows that $\rho(\cdot, A)$ is continuous.

Second part: $\operatorname{glb} \{ ||\beta - \alpha|| : \alpha \in A\} = 0$ iff there is a sequence of points $\alpha_1, \alpha_2, \ldots$ such that $\lim_{n \to \infty} ||\beta - \alpha_n|| = 0$; this is the definition of a convergent sequence with limit $\beta$, and such a sequence exists by definition iff $\beta \in \overline{A}$.

Note that we only need the metric-space axioms, not any special properties of normed linear spaces.