The numerator of the fraction on the right is an integer (because so are a, b, p, and q). The denominator is also an integer that stays constant for a given r. It follows that when approximating rationals, there is a fixed, minimum decrement for s: 1/q. We might start from a non-zero s1, but if we keep ramping up K, the system must reach the degenerate sn = 0 case after producing a finite number of inexact approximations. From that point on, s
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