Homotopic Equivalence of the set of all continous maps from $X$ to $\Bbb R^n$

Question

If we have the set of all continuous maps from $X$ to $\Bbb R^n$ denoted by $C(X,\Bbb R^n)$ and $\sim$ is the relation on this set, how could we prove that every two elements in $C(X,\Bbb R^n)$ is homotopic?

Answer 1

If $f,g: X \to \mathbb{R}^n$, then $f$ is homotopic to $g$ by the homotopy $H: X \times [0,1] \to \mathbb{R}^n$ defined by $H(x,t)= (1-t)f(x) + tg(x)$.

Clearly $H(-,0)=f$, $H(-,1) = g$ and $H$ is continuous.

All we need is that the codomain is convex to be able to do this. The domain $X$ plays no rôle of importance.