Suppose that I flip a coin repeatedly. After a certain number of flips, I tell you...

Suppose that I flip a coin repeatedly. After a certain number of flips, I tell you that less than 80% of my flips have been “heads”. I flip a few more times, and then tell you that now more than 80% of my (total) flips have been “heads”. Prove that there must have been a point where exactly 80% of my flips had been “heads”. (If we replace 80% with 40%, the statement is not true: if I flip “tails” twice, then “heads” twice, on the fourth flip I jump from 33.3 . . . % to 50% without ever attaining 40%. That is, a correct solution to this problem must use a specific property of 80%.)

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