Every time I look up this question I get a billion answers just showing Cantor's diagonal argument, or occasionally his nested-intervals argument.
But I don't think that actually proves that the reals are uncountable.
It just proves that if you have a recursively enumerable set $S$, and you arrange it into a sequence, then you can construct a real number in any given interval which is not a member of $S$.

So you can show that the reals are not recursively enumerable. But that doesn't mean that they're uncountable.
The set of computable numbers, for example, is countable, but not recursively enumerable.

Are there any more detailed proofs about the cardinality of the reals being greater than $\aleph_0$ ?