Arrangements in Hilbert's Hotel

Calculus Level 2

In Hilbert's Hotel, there are infinitely many rooms labeled 1,2,3,,1, 2, 3, \ldots, with one room for every natural number.

An infinite number of guests stay in the hotel: Mr. P1,Mr. P2,Mr. P3,, \text{Mr. }P_1,\, \text{Mr. }P_2,\, \text{Mr. }P_3, \ldots, with Mr. Pn \text{Mr. }P_n for every natural number nn.

Hilbert, the manager, realizes that every room is occupied by at least one guest.

Does that mean no two guests are in the same room?

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