OK, not using fractions, eh?
0.111111111111111111111111........
0.222222222222222222222222........
0.333333333333333333333333........
0.444444444444444444444444........
0.555555555555555555555555........
0.666666666666666666666666........
0.777777777777777777777777........
0.888888888888888888888888........
0.999999999999999999999999........
There.
0.999999999999999999999999....... = 1 because:
When you type 0.9999999.... for example, you are rounding 0.9 to 7 decimal places. And because of that, the '9' in the 8th decimal place will round up to '10' which means the 7th '9' will round up to 10, and so on... i.e.:
0.9999999....
= [sup]9[/sup]/[sub]10 [/sub]+ [sup]9[/sup]/[sub]100[/sub] + [sup]9[/sup]/[sub]1000[/sub] + [sup]9[/sup]/[sub]10000[/sub] + [sup]9[/sup]/[sub]100000[/sub] + [sup]9[/sup]/[sub]1000000[/sub] + [sup]9[/sup]/[sub]10000000[/sub]...
The 8th decimal will round the previous 9's up like so:
= [sup]9[/sup]/[sub]1[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]10[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub]...
= [sup]9[/sup]/[sub]1[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]10[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub]...
= [sup]9[/sup]/[sub]1[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]10[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub]...
= [sup]9[/sup]/[sub]1[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub] + [sup]10[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub]...
= [sup]9[/sup]/[sub]1[/sub][sub]0[/sub] + [sup]9[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub] + [sup]10[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub][sub]0[/sub]...
= [sup]9[/sup]/[sub]1[/sub][sub]0[/sub] + [sup]10[/sup]/[sub]1[/sub][sub]0[/sub][sub]0[/sub]...
= [sup]10[/sup]/[sub]1[/sub][sub]0[/sub]
= 1
YAY! It's using fractions, but representing the decimal places, not as a single fraction.