Hilbert's hotel

The Grand Infinite in Goa has a sign outside: Rooms 1, 2, 3, 4… and so on, forever. On the night of the big festival, every single room is taken.

At midnight, a tired traveller called Sam walks in. The receptionist, Mrs D'Souza, smiles. She makes an announcement: "Would every guest please move from their room to the next room up?" The guest in room 1 moves to 2, room 2 moves to 3, and so on forever. Nobody is left without a room, and room 1 is now free for Sam.

At one in the morning, an infinitely long bus pulls up, with infinitely many passengers. The receptionist doesn't even blink. "Every guest, please move to the room whose number is double yours." Room 1 goes to 2, room 3 to 6, room 10 to 20.

Now every odd-numbered room, 1, 3, 5, 7…, is empty, and there are infinitely many of those. The whole bus checks in. The hotel was full, and now it holds infinitely more guests.