0.999.../Proof by limit manipulation

Assumptions

  • Definition from sequences
  • Term-by-term operations on sequences
  • The limit of a geometric sequence

Proof

0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,
Last modified on 27 November 2011, at 21:30