Let $U$ be open in $\Bbb R^n$ and $\alpha$ a path of $U$. Show that $\alpha$ is homotopic to some polygonal chain (polygonal path). 
Let $U$ be open in $\Bbb R^n$ and $\alpha$ a path of $U$. Show that $\alpha$ is homotopic to some polygonal chain.

Definition first. A path $\beta:I \to \Bbb R^n$ is polygonal chain if it's of the form $\beta(s)=a_js+b_j$ for $s_{j-1}\le s \le s_j$ and $1 \le j \le N.$ I.e it's just a "chain" of line segments.
So if $m$ is the polygonal chain, then I need to find a map $H:I^2 \to \Bbb R^n$ such that $H(x,0)=\alpha(x), H(x,1)=m(x), H(0,t)=\alpha(0)=m(0)$ and $H(1,t)=\alpha(1)=m(1).$
I tried to see why the linear homotopy doesn't work so if I let $H(x,t)=(1-t)\alpha(x)+tm(x)$ I have the first two conditions satisfied $$\begin{align} H(x,0)&=\alpha(x) \\ H(x,1) &= m(x) \end{align}$$ but I'll end up in trouble with $$\begin{align} H(0,t)&=(1-t)\alpha(0) + tm(0) \ne \alpha(0) =m(0)\\ H(1,t) &= (1-t)\alpha(1) + tm(1) \ne \alpha(1) =m(1). \end{align}$$
So I would somehow need to pick $m$ such that it starts and ends at $\alpha(0)$ and $\alpha(1)$ respectively? Is that what is happening here?
 A: So this is the idea behind the proof:

You have the large open set $U$, here containing a hole which might be a possible obstruction to any homotopy. The path $\alpha$ is in black. Now the idea is to cover this path using a finite number of open, convex sets, each of which is contained in $U$. The small discs in the image. A hint to prove that such a covering exists: The path is compact (which you need to prove, of course).
Now the finite number of convex sets has overlaps. The idea is to take points of the path which are in these overlaps and to connect these points using a line (green). You can now show that the segment of $\alpha$ connecting the chosen points is homotopic to the line connecting these points. You can prove this in general: given a convex set $C$ and two points $a,b\in C$, any two continuous paths connecting the two points are homotopic.
Now you just need the following: If the paths $\alpha_1,\dots,\alpha_n$ are homotopic to the paths $\beta_i,\dots,\beta_n$, then $\alpha_1+\dots+\alpha_n$ is homotopic to $\beta_1+\dots+\beta_n$, where "$+$" means concatenation of paths. Now since $\alpha$ is just the concatenation of the smaller segments, all of which are homotopic to a corresponding line segment, it is then homotopic to the concatenation of these line segments - which is a polygonal chain.
Some details you will have to take care of: The convex sets (discs would be the conventient choice) might have overlaps with multiple other discs. Also, multiple segments of $\alpha$ might go through some of the discs. How do you choose the "correct" points of $\alpha$ from within the overlap in order to construct the polygonal chain?
