Distance from a Point to a Nonempty Subset of a Metric Space

Let $(X,d)$ be a metric space, let $A\subseteq X$ be nonempty, and let $x\in X$. Let

$$S_{x,A}=\{t\in\mathbb{R}:\ t=d(x,a) \text{ for some } a\in A\},$$

where $\mathbb{R}$ denotes the real numbers. Then $S_{x,A}$ is nonempty because $A$ is, and $0$ is a lower bound for $S_{x,A}$ by condition 1 of the definition of a metric, so the greatest lower bound of $S_{x,A}$ exists by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below.

The distance from $x$ to $A$ in $(X,d)$ is the real number

$$\operatorname{dist}_d(x,A)=\inf S_{x,A}.$$