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 $$X$$.