Hilbert's hotel: why can't I repeat it infinitely many times? I was wondering about the following:


  Suppose a new guest arrives and wishes to be accommodated in the
  hotel. We can (simultaneously) move the guest currently in room 1 to
  room 2, the guest currently in room 2 to room 3, and so on, moving
  every guest from his current room n to room n+1. After this, room 1 is
  empty and the new guest can be moved into that room. By repeating this
  procedure, it is possible to make room for any finite number of new
  guests.


However, if we have an infinite amount of guests, why can't we just say that each guest follows this procedure? Everyone would have to relocate infinitely many times, but what's the problem?
Keep in mind that Hilbert's Hotel is really just an analogy for analyzing countable and uncountable sets, i.e., deciding whether we can construct a bijection from $\mathbb{N}$ to a given set.

If you insist on staying within the analogy, here's the issue with telling everyone to move "infinitely many times." As a hotelier, you need to have a list of who is each room. Similarly, each of your (very accomodating) guests, needs a specific room number to move to. If a guest is in room #1, telling them to move to room #2 (or room #37 or room #123094871230948172 or ...) is a well defined operation. While it may be a long walk, they can get to that room in finite time and know exactly where they are headed. 

On the other hand, telling the guest in room #1 to keep walking down the hall until they seem a room that has a number larger than any natural number on the door is ill-defined. There is no way that they could possibly reach this room by walking down the hallway. In effect, you have told them "I don't actually have a room for you, but you are welcome to spend the rest of your life walking down the hallway trying to find one!" This sort of behavior by hotel management is generally frowned upon and leads to poor online reviews of the establishment.
You can accomodate a countably infinite number of new guests in Hilbert's Hotel. Just have each current guest, who is in room number i, move to room 2i. Then all the odd numbered rooms are open and the new guests can move in, and everybody only has to move once.