I have got this exercise: If space $Y$ is a convex subspace of $\mathbb{R}^n$ and $X$ --- any topological space, then any two continuous functions $f,g : X \rightarrow Y$ are homotopic.
My proof: If Y is a convex space, then any two points $x,y$ of this space $Y$ can be connected by some line segment $[x,y]$. The images of $X$: $f(X)$, $g(X)\subset Y$ are locally equivalent to a structure of preimage $X$ (because of continuity). So, the images should have similar properties of connectedness: a similar number of connected components, number of $n$-dimensional holes in them etc.
I can immediately connect any point $f(x) \in f(X)$ with $g(x) \in g(X)$, where $x$ --- is a unique point in domain $X$, by line segment $[f(x), g(x)]$. Any two line segments in $\mathbb{R}^n$ are homeomorphic, so I can change notation of segment from $[f(x), g(x)]$ to $[0,1]$.
Homotopy $H$ should be looking like that: $H: f(X) \times [0,1] \rightarrow g(X)$, where any $H(f(x),t)$ is just an a point $t$ of segment $[0,1]$, that connects points $f(x)$ and $g(x)$.
Can you check please my proof? And send me comments, I will be grateful for that.