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If this pattern cannot be continued infinitely, what is its limit?
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Any limit, it doesn't matter. As long as you have to take it out infinitely long, then there doesn't exist a pair of integers that form a fraction that equals the irrational number.
Let's turn it around. Take pi. Go ahead and form your "infinitely long" fraction that is equal to pi. Now tell me what the numerator and denominator are. If they are integers, then you've proven your point.
The problem is they can't be integers, because they will both be infinitely long without any periodicity.