I would like to prove this using max$\{x_1,x_2\}=\frac{x_1+x_2}{2}+\frac{|x_1-x_2|}{2}$ and by induction
So the idea here is that max $\{x_1,x_2,...,x_n\}=max\{max\{x_1,x_2\},max\{x_2,x_3\}...max\{x_{k-1}x_k\}\}$
It's recursive and therefore is very, very messy
Proof
Basis (n=2): max$\{x_1,x_2\}=\frac{x_1+x_2}{2}+\frac{|x_1-x_2|}{2}$ is true
Assume for some $k \in \mathbb{N}$, max$\{x_1,x_2,...,x_k\}=max\{\frac{x_1+x_2}{2}+\frac{|x_1-x_2|}{2},\frac{x_2+x_3}{2}+\frac{|x_2-x_3|}{2},...,\frac{x_{k-1}+x_k}{2}+\frac{|x_{k-1}-x_k|}{2}\}$ is true
Inductive: Show this holds for k+1
But... ?
This is where I get stuck, though. Plus I'm not even sure that my algebra is right for my "assume k" case because, while true, doesn't seem to prepare us adequately for the induction
I feel like you can't prove this without taking up a very large amount of space, since it is recursive