I stumbled upon this proof of $\pi$ being rational (coincidentally, it's Pi Day). Of course I know that $\pi$ is irrational and there have been multiple proofs of this, but I can't seem to see a flaw in the following proof that I found here. I'm assuming it will be blatantly obvious to people here, so I was hoping someone could point it out. Thanks.
We will prove that pi is, in fact, a rational number, by induction on the number of decimal places, N, to which it is approximated. For small values of N, say 0, 1, 2, 3, and 4, this is the case as 3, 3.1, 3.14, 3.142, and 3.1416 are, in fact, rational numbers. To prove the rationality of pi by induction, assume that an N-digit approximation of pi is rational. This number can be expressed as the fraction M/(10^N). Multiplying our approximation to pi, with N digits to the right of the decimal place, by (10^N) yields the integer M. Adding the next significant digit to pi can be said to involve multiplying both numerator and denominator by 10 and adding a number between between -5 and +5 (approximation) to the numerator. Since both (10^(N+1)) and (M*10+A) for A between -5 and 5 are integers, the (N+1)-digit approximation of pi is also rational. One can also see that adding one digit to the decimal representation of a rational number, without loss of generality, does not make an irrational number. Therefore, by induction on the number of decimal places, pi is rational. Q.E.D.