This algorithm appears and is analyzed in Curtis, Darts and hoopla board design. Curtis actually considers two problems:

1. Dartboard design: Given a sequence of numbers, find a circular arrangement $a_1,\ldots,a_n$ maximizing $\sum_{i=1}^n |a_i-a_{i+1}|^q$ for some $q \geq 1$ (where $a_{n+1} = a_1$).

2. Hoopla board design: Given a sequence of numbers, find a permutation $a_1,\ldots,a_n$ maximizing $\sum_{i=1}^{n-1} |a_i-a_{i+1}|^q$ for some $q \geq 1$.

Your problem is the hoopla board design with $q = 1$. You can find a complete proof of the greedy algorithm in Curtis' paper. It's trivial, though also not too complicated.

This algorithm appears and is analyzed in Curtis, Darts and hoopla board design. Curtis actually considers two problems:

1. Dartboard design: Given a sequence of numbers, find a circular arrangement $a_1,\ldots,a_n$ maximizing $\sum_{i=1}^n |a_i-a_{i+1}|^q$ for some $q \geq 1$ (where $a_{n+1} = a_1$).

2. Hoopla board design: Given a sequence of numbers, find a permutation $a_1,\ldots,a_n$ maximizing $\sum_{i=1}^{n-1} |a_i-a_{i+1}|^q$ for some $q \geq 1$.

Your problem is the hoopla board design with $q = 1$. You can find a complete proof of the greedy algorithm in Curtis' paper. It's not trivial, though also not too complicated.