Hotel Infinity (Part 1)

Calculus Level 3

You are the hard-working manager at Hotel Infinity, an inn with an infinite number of rooms, each with a room number \(n\), where \(n\) is a natural number. No matter how crowded the hotel is, you can always make room for one more guest: you move the person from room 1 to room 2, the person from room 2 to room 3, the person from room 3 to room 4, and so forth. After you've moved all of the guests to their new rooms, the new guest can take room 1. Unfortunately, as soon as your about to go off-duty, a group made up of an infinite number of people arrive at the hotel. If you already have an infinite number of guests, what's the most efficient way to accommodate the newcomers so that every room in the hotel is occupied?

Clarification: The number of rooms is countably infinite.

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