From Riemannian geometry, the distance between two points $x$ and $y$ on a Riemmanian manifold $M$ is given (abstractly) by
$d(x,y)= \inf \lbrace \ell (\gamma) : \gamma \in \Omega_{x,y} \rbrace$, (1)
where $\Omega_{x,y} = \lbrace \gamma : [0,1] \rightarrow M : \gamma \ \text{piecewise smooth and} \ \gamma(0)=x, \gamma(1)=y \rbrace$.
For the case $M=\mathbb{R}$, the Euclidean distance between two points $x$ and $y$ is $|x-y|$ (2).
I would like to see that these two notions coincide, i.e. that one can obtain (2) from (1) by a direct computation. Several books state that this is the case, but I can't find a formal proof. If anyone know of a reference, I would be grateful.
Let $\gamma\colon [0,1]\to\Bbb R$ be a piecewise smooth map with $\gamma(0)=x$ and $\gamma(1)=y$. For simplicity here, I'll assume it's actually a smooth map, but you can fix up the argument for the general case. $$\ell(\gamma) = \int_0^1 |\gamma'(t)|\,dt \ge \left|\int_0^1 \gamma'(t)\,dt\right| = |y-x|,$$ as desired. Note this proof works identically in $\Bbb R^n$.