Here is a (long and convoluted) way to multiply 0.999... by 0.999...:
**0.999...
x 0.999...
-------------
I will break this into separate components that will all be added back together at the end. Since 0.999... can be expressed as the geometric series "0.9 + 0.09 + 0.009 + 0.0009...
Allow me to demonstrate on how the multiplication can be done.
Now, traditionally when multiplying the following (the *s are used for space-filling, please adjust fonts or copy-paste into a simple text editor to line up characters) (I'm a newbie and can't figure out the LaTeX codes that would...