I'm not familiar with the construction of the reals, but somewhere in there you'd have to say what 0.999... means for it to be equal to 1. I'd probably just de

I'm not familiar with the construction of the reals, but somewhere in there you'd have to say what 0.999... means for it to be equal to 1. I'd probably just define it as an infinite series, where it's straight forward to show it is 1. But on second thought, kids run into decimals before constructing the reals, I'd imagine you could define rational numbers and repeating decimals and end up with a simpler argument.