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It's true that that there are numbers that aren't rational... let's say x is such a number. Now we take two integers, a and b where a is the integer if x is rounded up, and b is the integer if x is rounded down.

Forming their arithmetic average, we will get a value y that is closer to x than either a or b. If x is smaller than y, we form the average of b and y, and if it's greater, we use a and y...

Keeping on like this we can come arbitrary close to x, expressed with integrals. So there must be infinitely many rational numbers just between a and b. How can it then come that we still can't write x as division with two integers?