(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!
... 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,
0.3
0.33
0.333
0.3333
0.33333
...
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.
Signed:
The Everything2 Mathematics Police
"Making E2 a safer place for mathematics undergraduates"