Archimedes used areas of regular polygons to approximate pi. He calculated both inner and outer polygons and realized that more sides yielded closer results to each other. There's surviving proof that he did this through at least the 96 side polygon.
Did Archimedes or any other ancient actually prove that the inner and outer techniques would converge? It's pretty easy to show that the ratio of an inner to outer line segment approximation is the cosine of half the angle of one side of the polygon, and that the ratio of an area approximation would be cosine squared. As such as the number of sides increase, the two techniques converge.
Did anybody from the distant past mention an argument similar to this and drive that point home? Or did they merely realize that each subsequent iteration of inner and outer was getting closer to each other?
Archimedes proved (up to a significant rewording) that for the unit circle, for any value less than $\pi$, he could find an inscribed polygon whose area was greater than that value, and that for any value greater than $\pi$, he could find a circumscribed polygon whose area was less than $\pi$.
With the exception of showing that the perimeter of the circumscribed polygon must be greater than the circumference of the circle, his proof is quite rigorous. (The exception only occurs because Archimedes did not have a rigorous definition of arclengh, and so could not prove this one part rigorously.)
In modern parlance, it amounts to a proof that there are two sequences of polygons whose areas converge to $\pi$. One from below, and the other from above. His estimation for $\pi$ comes from polygons that can be shown to be on those sequences.