Let $x = 0.9999\cdots$ which, by definition is $$ x = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots $$ This series clearly converges. Then, $$ 10x = 9 + \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots = 9 + x \; \Rightarrow \; 9x = 9\; \Rightarrow \; x = 1. $$