Projection of space curve shortens Let $C$ be a rectifiable, open curve in $\mathbb{R}^3$,
and let $|C|$ be its length.
Orthogonally project $C$ to a plane $\Pi$ (e.g., the $xy$-plane).
Call the projected curve $C_{\perp}$, and its length $|C_{\perp}|$.
I would like to claim $|C_{\perp}| \le |C|$.
I would appreciate either a simple proof, or a reference.
This may be so well-known that it is hard to cite a reference.
(I only need it in $\mathbb{R}^3$, but it should hold in any dimension.)

If you are dealing with rectifible curves, you are taking polygons with vertices
on $C$ and looking at the limit at the lengths of the polygons as the points
become closer. But if you project a line segment to $\{z=0\}$ its length cannot increase,
so the projected polygons are no longer than the original.
