This is a pretty nice proof, however, my smooth brain tells me that Archimedes's method of exhaustion to "approximate" pi showed us that this process can go on

This is a pretty nice proof, however, my smooth brain tells me that Archimedes's method of exhaustion to "approximate" pi showed us that this process can go on and on forever because there is no "intersection" point on a circle that stops the circumscribing...so there will always be a new set of fractions that approximate pi that is better than ever...hence pi has to be irrational... Despite its obvious flaw of not having a rigorous proof but it is quite intuitive and easy to understand, and no need for definition of trigonometry functions, just need to have the concept of infinitesimal. Edit: I just found out that pi can be expressed as infinite series of continued fractions, hence proving its irrationality. Found it interesting because it is almost exactly same as I thought. I think of that "proof" when I was in high school.