I was reading about (the famous Greek philosopher and mathematician) Archimedes' "Method of Exhaustion":
The "Method of Exhaustion" is used to approximate the area of different shapes. For example, suppose you have a circle. The "Method of Exhaustion" requires you to:
Take a "n" sided polygon (assuming that you know the area of this polygon) and inscribe this polygon (i.e. draw inside) inside the circle. You will notice that the area of the circle is the difference between the area of the "n" sided polygon and some "remaining area". Conversely, the "error" ("remaining area") is the difference between the (unknown) area of the circle and the "n" sided polygon.
Next, you repeat this process with an "n+1" sided polygon : Once you inscribe a "n+1" sided polygon within the circle, you will notice that the "remaining area" (i.e. "error) will appear smaller when compared to the previous step with the "n" sided polygon.
As you increase the number of sides corresponding to the polygon that you inscribe within the circle, you will notice that "remaining area" will keep reducing : Theoretically, a "infinite sided polygon" should result in "no remaining area" (i.e. "zero error"), as a circle can be thought of as a "infinite sided polygon".
You can also place a "bound" on the true area of the circle: Given that you have drawn an "n" sided polygon inside the circle, if you were to also draw a larger "n" sided polygon outside the circle - the difference between these two areas would "bound" the true area of the circle.
I have the following question: Either in Archimedes' time or in Modern times - can we "prove" (e.g. mathematical induction) this extremely obvious and evident fact : As the number of sides of the polygon increases, the "remaining area" of the circle CAN NOT increase?
Or this fact considered unprovable and its validity is ascertained using the Mathematical Axioms of Geometry?
Thanks!