Hilbert's Hotel

I recently stumbled across a thought experiment in mathematics, where the counterintuitive nature of infinity is portrayed. There is a hotel with an infinite number of rooms, all of which are occupied with a guest. There is a paradox upon the arrival of a new guest, the hotel is still shown to be able to accomodate this guest, despite it being previously coined as full.

See: this link for more details on the full experimental thought and the proofs alongside.