Hilbert's Hotel: Accommodating Infinite Guests
You manage a hotel with infinitely many rooms, numbered
, 2, 3, \ldots$, and every room is currently occupied.
**Part 1:** A single new guest arrives. How do you make room for them without kicking anyone out?
**Part 2:** Now an infinite busload of new guests arrives -- countably infinitely many people, each needing a room. All your rooms are still full. How do you accommodate all of them?
**Part 3:** Now infinitely many buses arrive, each carrying infinitely many passengers. Every room is still occupied. How do you fit everyone in?
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