Recently, I learnt in my analysis class the proof of the uncountability of the reals via the [Nested Interval Theorem][1] ([Wayback Machine](http://web.archive.org/web/20180402194030/http://personal.bgsu.edu/~carother/cantor/Nested.html)). At first, I was excited to see a 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 use Cantor's diagonal argument.

Is my belief justified? 

Thank you.


 


  [1]: http://personal.bgsu.edu/~carother/cantor/Nested.html