Here is my question while reading a proof.

For a compact set [tex] K [/tex] in a separable metrizable spce [tex] (E,\rho) [/tex] and a continuous function [tex] t \mapsto f(t) [/tex], if we define

[tex] D_{K} = \inf \{ t \geq 0 \; : \; f(t) \in K \}[/tex]

then, [tex] D_{K} \leq t [/tex] if and only if [tex] \inf\{ \rho(f(q),K) : q \in \mathbb{Q} \cap [0,t] \}[/tex] = 0

May someone shed some light on this? I do not understand it. Thanks very much.

Wayne