system
Given the following post, respond as if you're commenting on Reddit:
human
Is there any non graphic proof that when n appoaches infinity, |(x^n)| + |(y^n)| = r is a square?

I've been playing with online cartesian drawing tools. When n=1, it is a rotated square at half PI. Then it will transform into a circle while rising n slowly towards 2.

Then, an interesting thing began. Any increase in n will make it more square-y but will never become a complete square.

Will it became a true square when n reaches infinity? What is the proof?
gpt
Let's use r=1. You can ask for which x,y does |x^(n)| + |y^(n)| converge to 1. Let's say x,y>0, r=1 and x=cy for 0<c<1, the other cases can be treated using symmetry or they are trivial. Then |x^(n)| + |y^(n)| = y^n (1+c^(n)). For y<1 this converges to zero, for y>1 this diverges, for y=1 this converges to 1.

If you use a different r then you never get convergence to r of a point, but you can still show that you get the closest to r at this square.