So I need to show for an open subset $U \subset \Bbb{R}^n$ any path
$$\gamma:[0,1] \to \Bbb{R}^n$$
such that $\gamma(0)=a, \gamma(1)=b$ is homotopic to a polygonal path from $a$ to $b$.
Now for definition of polygonal path do we use the following:
A path $\alpha$ is $\textit{polygonal}$ if there exists a partition $0 = t_0 < t_1 <... < t_k=1$ such that $\alpha$ maps $[t_{j-1},t_j]$ onto the straight line from $\alpha(t_{j-1})$ to $\alpha(t_j)$. i.e.,
$$\alpha([t_{j-1},t_j])=(1-t)(\alpha(t_{j-1}))+t(\alpha(t_j)).$$
Now I know that $\gamma([0,1])$ is compact so it can be covered by finitely many open balls that are convex such that their intersections are non-empty. So could I partition my open sets in a way that each of them contains an $\alpha[t_{j-1},t_j]$ then use the straight line homotopy from my path $\gamma$ that lies in that same ball to deform it into a line segment? Sorry just having a tough time making this rigorous. Please go easy on me, im new to algebraic topology. I wanna say it has something to do with the distance between $\gamma$ and the boundary of $U$, call this distance $d$. Then choose my partition in such a way that
$$0<t_0 <...<t_k=1$$
is such that
$$\gamma(t)-\gamma(t_j)$$
has norm less than $d$ for $t$ between $t_j$ and $t_{j+1}$.