TITLE: Distance of two points in $\mathbb{R}^N$ $\leq$ length of curve
QUESTION [5 upvotes]: Let $\gamma$ be a $C^1$-curve and $x,y \in \mathbb{R}^N$ with $\gamma: [0,1] \longrightarrow \mathbb{R}^N, \gamma(0)=x, \gamma(1)=y$.
My intuition tells me that $||x-y|| \leq \ell(\gamma)$, where $\ell$ is the length of a curve. Is this true and how to show it?

REPLY [6 votes]: This can be easily seen with a straightforward calculation:
$$\| y - x \| = \| \gamma (1) - \gamma (0)\| = \left \| \int_0^1 \dot{\gamma}(t)\ \mathrm dt \right  \| \leq \int_0^1 \|\dot{\gamma}(t)\| \ \mathrm d t = \ell(\gamma). $$