At some point in his career, Mario Mendoza's batting average was above 0.200. Sometime later, it was below 0.200. Prove that, at some point, he was batting exactly 0.200.
For those unfamiliar with baseball: players have a series of "at-bats" (attempted hits) which either succeed, or don't. At any point, a player's batting average is the fraction of their at-bats so far which were hits. This number is usually rounded to 3 decimals, but for the purposes of this puzzle, consider all fractions to be exact.
For those familiar with baseball: the point where he is batting 0.200 might occur during a game, so assume that a player's batting average is updated after every at-bat.
Unnecessary fun fact: Mario Mendoza was a real player, whose batting average tended to be around 0.200, an embarrassingly low rate. This is pretty led his teammates to joke that anyone who batted below 0.200 was "below the Mendoza line."