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 $\epsilon >0$ and a sequence of maps $f=f_{0},f_{1},f_{2},\ldots ,f_{n}=g$ such that for every $x\in X$:

$d(f_{i}(x),f_{i-i}(x))<\epsilon$

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