Real number Infinite Hotel

You see a hotel with a infinite amount of rooms. The Hilbert Hotel is a hotel with a infinite amount of floors. There is a elevator that serves infinite floors, from the lobby to level $\aleph _{0}$ . (This is the highest floor of the hotel.)

Hilbert Hotel

The Hilbert Hotel is a grand hotel with an infinite amount of rooms with a infinite amount of guests, and a extremely tired night manager.

If a finite of guests want to check in

One guest

The manager will ask the guests to move from number $n$ to $n+1$ .

The guest in room number 1 will be asked to move into room 2.

The guest in room number 2 will be asked to move into room 3. etc.....

2 guests

The manager will tell the person in Room 1 to move to Room 3.

Example:

$n$ to $n+2$

Etc...

If a infinite number of guests come in

You see a infinitely long bus carrying a countably infinite amount of passengers. You don't know what to do, but you are smart. You ask the person in number 1 to room 2. You ask the person in Room 2 to move to Room 4.

$n$ to $n\cdot 2$

Then, the unthinkable happens.

You see a infinite amount of infinitely long buses carrying a countable infinite passengers.

How can you even do this?

You are smart, remember. "Remember, there are infinite prime numbers." by Georg Cantor. In order for this to work, The first prime, 2 , you need to raise 2 to the power of your current room number. $n$ to Failed to parse (syntax error): {\displaystyle n^{$\: your current room number$} }