Pi contains pi as showing by proof by induction:
- 1. pi contains pi as one significant digit in a finite representation ("3": 3.141592653....)
- 2. assume pi contains pi as n significant digits in a finite representation of pi. Pi must also contain pi as n+1 significant digits as the number of digits of pi is stretched to infinity.
Stated another way, pi includes every number that has a finite representation. Thus pi includes itself in all of its increasingly precise representations, without limit, and therefore contains pi within a vanishingly small margin of error.