Wb/yue/0.999.../用極限證明

證明

0.999 . . . = \lim_{n \to \infty} 0 . \underset{n}{\underset{⏟}{99 . . . 9}} = \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{9}{1 0^{k}} = \lim_{n \to \infty} (1 - \frac{1}{1 0^{n}}) = 1 - \lim_{n \to \infty} \frac{1}{1 0^{n}} = 1 .