>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each one digit is not in the list. Then it's possible to fix the details and use the real numbers instead of the infinite strings of digits.

What am I missing?

"0.765653625367523765..." could be assigned the transfinite natural number beginning "1765653625367523765..."

"0.000073468763478..." could be assigned the transfinite natural number beginning "1000073468763478..."

Transfinite natural numbers must exist otherwise you do not have an infinite set.