I recently started studying set theory and am having quite a bit of difficulty accepting Cantor's diagonal proof for the uncountability of the reals. I also saw a topological proof via nested sets for uncountability which still does not satisfy me completely, given that just like the diagonal it relies on a never ending process. In fact, the nested sets proof sounds very much like the diagonalization proof to me.
Do all proofs of the uncountability of the reals involve diagonalization? Are there any other proofs I can look at to understand? I couldn't find any on searching stack exchange. Thanks.