Suppose I have two points $\alpha,\beta \in \Bbb{R}^n$. Define $ X=\{\gamma \in C^1([0,1] , \Bbb{R}^n),\ \gamma(0)=\alpha,\gamma(1)=\beta ,0 <|\gamma'|
In my case I can prove that $\gamma([0,1])\subset [\alpha,\beta]$ (the line segment joining $\alpha$ and $\beta$). My question is:
Can I find a sequence of paths $\theta_n \in C^1([0,1],[\alpha,\beta])$ for which $|\theta_n'| \leq K$ and $(\theta_n)$ converges uniformly (or maybe pointwise) to $\gamma$?
My intuition says that if I can approximate $\gamma$ with a sequence of $C^1$ paths joining $\alpha$ and $\beta$ then it is sufficiently regular such that I can approximate $\gamma$ with $C^1$ curves with $\theta_n([0,1])\subset \gamma([0,1])$.
One thought was to use projections of $\gamma_n$ on the segment $[\alpha,\beta]$, but I think that in some cases the projection can destroy differentiability.