Hey this is pretty cool. You said “the proof plays out differently for other common repeating decimals” but I guess it depends on what you mean by “differently.” If you mean it doesn't equal 1, then yes, it's different. When you subtract 0.9999… from 10 * 0.9999… , you are essentially adding one more decimal point (*10) to the first term in the difference. Or, if you'd rather consider it: you're always one iteration ahead, just enough to evaluate the limit.
But this is why it's really the same for other repeating decimals like
:
0.999… approaches 1, which is the answer you get
0.333… approaches
which is the answer you get.
P.S. I like your monobook theme, that's why I'm here. Nice site too.