(Originally written to explain why the "trivial" claim that π must be rational, as it has a converging sequence of rationals, doesn't hold water. This still has merit, so I'm keeping it: 2^.5 = 2 shows a similarly misguided approach to mixing up proof by induction with what happens at the limit. Not everything is continuous and not all sets are closed!

π is rational

... by induction on the number of decimal places, any k-digit approximation to pi is rational.

Which means very little, seeing as any real number has arbitrarily close rational approximations.

One might as well claim 1/3 has a terminating decimal expansion. After all,

all have terminating decimal expansions, and they're converging approximations to 1/3.

With apologies to fustflum, whom I trust realises this is not a real proof.

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