0.1999... and 0.2000 are the same? From Sipser's Introduction to the Theory of Computation:

A slight problem arises because certain numbers, such as 0.1999... and 0.2000..., are equal even though their decimal representations are different.

Can someone explain? Because that's news to me.

$$0.1999999...=\frac{1}{10}+\frac{9}{100}+\frac{9}{1000}+...$$
$$=\frac{1}{10}+\frac{9}{100}(1+\frac{1}{10}+\frac{1}{100}...)=\frac{1}{10}+\frac{9}{100}\frac{1}{(1-\frac{1}{10})}=\frac{1}{10}+\frac{1}{10}=\frac{1}{5}=0.2$$
