Packing and covering numbers with explicit radius conventions

Suppose a cover of $\mathcal{F}$ by $k<\mathrm{\infty }$ closed balls of radius $ϵ\mathrm{/}2$ is given. Two points in the same ball have distance at most $ϵ$ by the triangle inequality, so an $ϵ$-separated subfamily contains at most one point from each ball. Hence every finite separated subfamily has size at most $k$, and $D(ϵ,\mathcal{F},d)\le k$. Taking the least such $k$ proves $D(ϵ,\mathcal{F},d)\le N(ϵ\mathrm{/}2,\mathcal{F},d)$; if no finite cover exists the inequality is automatic.

If $D(ϵ,\mathcal{F},d)=+\mathrm{\infty }$, then $N(ϵ,\mathcal{F},d)\le D(ϵ,\mathcal{F},d)$ is immediate. Otherwise the finite separated cardinalities are a nonempty bounded subset of the natural numbers, so a largest size is attained by some finite separated subfamily $G\subseteq \mathcal{F}$.

Maximality of $G$ means each $f\in \mathcal{F}$ has $d(f,g)\le ϵ$ for some $g\in G$; if all distances were greater, adding $f$ would make a larger separated subfamily. Thus the balls centered at $G$ with radius $ϵ$ cover $\mathcal{F}$, so $N(ϵ,\mathcal{F},d)\le \mathrm{\mid }G\mathrm{\mid }=D(ϵ,\mathcal{F},d)$.