...999$...999$ is, I assume, an infinite string of 9s$9$s, thus is it infinity. 0.1 * infinity $0.1 \cdot \infty$ is still infinity$\infty$. On line 3, you then say 0.1x - x$0.1x - x$, but since both terms are infinity, what you are really saying is infinity - infinity$\infty- \infty$, which is an undefined operation, so the rest doesn't hold. If you abuse infinity like it's a number (it's not) you can get all sorts of contradictory results. For example:
1 + infinity = infinity$1 + \infty= \infty$
1 = infinity - infinity = 0$1 = \infty- \infty= 0$
Thus, 1 = 0$1 = 0$
Which is obviously wrong.
