Homotopic maps are close

Statement

Suppose $X$ is a compact space and $(Y,d)$ is a metric space. Suppose $f$ and $g$ are homotopic maps from $X$ to $Y$. Then, there exists an $ϵ>0$ and a sequence of maps $f=f_0,f_1,f_2,\dots ,f_n=g$ such that for every $x\in X$:

$d(f_i(x),f_{i-i}(x))<ϵ$

A related fact is that close maps are homotopic: the condition of compactness is now on $Y$ instead of on $X$.