Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see variant proof (as it did not use the diagonal argument explicitly). However, as time passed, I began to see that the proof was just the old one veiled under new terminology. So, till now I believe that any proof of the uncountability of the reals must uses Cantor'sdiagonal argument.

Is my belief justified?

Thank you.