Comment by boxy310

Yes, it's quite clever. An equivalent proof is dividing 0.999... by 9 using long division, which comes out to 0.111... which is equal to 1/9. Now use fraction notation and it simplifies to 9/9 = 1. Not quite as robust as the limit-based proofs but it's a quick answer and gets to the heart of the issue of repeating notation not capturing the whole picture.