A more concise and intuitive demonstration that 0.9999...=1

[InvisibleBison]

Start with the number 1.1111....repeating
Subtract 1 (leaving you with 0.1111...repeating)
add 0.9999....repeating

and you see immediately that it gives you back the original number

I don't think this is a good proof. It looks to me that you're assuming what you're trying to prove on line 3. If .9 repeating doesn't equal 1, then adding it wouldn't result in the original number.