I had a curious thought; given two distinct points $x,y\in\mathbb{R}^n$ for any $n\geq 2$, is every path connecting them, $$\gamma : [0,1] \to \mathbb{R}^n \mbox{ such that } \gamma(0)=x, \gamma(1)=y, \gamma \mbox{ continuous} $$ homotopic to the straight line connecting them, $$ \ell : [0,1] \to \mathbb{R}^n \mbox{ such that } \ell(t)= (1-t)x + ty ?$$
In other words, given any such path $\gamma$, does there always exist a continuous mapping $H:[0,1]\times[0,1] \to \mathbb{R}^n$ such that $$ H(t,0)=\gamma(t), H(t,1)=\ell(t)? $$
My suspicion is that the answer is yes, since paths are essentially one dimensional subsets of $\mathbb{R}^n$. The only defective case I can imagine is one where the path forms a loop and self intersects. But I think the crucial step of the solution to that case is picking a point on the loop, and showing the path before that point is homotopic to a line segment, and same with the path after that point.
The issue is I don't have enough experience to prove this rigorously, and also I could be missing some simple counterexamples. Any ideas?