Can rectifiable curves be approximated by smooth curves, controlling the length ratio? – mathoverflow.net

Let $\gamma:[0,1]\to\mathbb{R}^N$ be a rectifiable curve and $\ell(\gamma)$ be its length. Also denote $\Gamma=\{\gamma(t)|\ t\in [0,1]\}$ the image of $\gamma$. Is it true that for each ...