According to Wikipedia:
Although a room can be found for any finite number of nested infinities of people, the same is not always true for an infinite number of layers, even if a finite number of elements exists at each layer.
The set of real numbers, and the set of guests in this example, is uncountably infinite.
I understand Cantor's diagonal arguments, and thus why we can't fit uncountably many guests into a hotel with countably many rooms. However, I don't understand how countably many layers of nesting, with finitely many elements in each layer, ends up with uncountably many elements.
How can we prove that the number of guests in this example (countably many layers of nesting, finite elements on each layer) is uncountable?
(Inspired by this question.)