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

An infinite number of guests stay in the hotel: $\text{Mr. }P_1,\, \text{Mr. }P_2,\, \text{Mr. }P_3, \ldots,$ with $\text{Mr. }P_n$ for every natural number $n$.

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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