Suppose that Hilbert’s Grand Hotel is fully occupied on the day the hotel expands to a second building which also contains a countably infinite number of rooms. Show that the current guests can be spread out to fill every room of the two buildings of the hotel.

Step 1:

We need to show that the current guests can be spread out to fill every room of the two buildings of the hotel.

Step 2:

The definition of Hilbert’s Grand Hotel:

Hilbert’s Hotel is an Experiment which illustrates a counterintuitive property of infinite sets. therefore It is accomplished that a fully occupied hotel with infinitely many rooms may still accommodate additional guest and that this process may be repeated.