I understand how in the infinite hotel 'paradox' moving every person in room $n$ to room $n+1$, and then putting the new quests in room $1$, generates a new space in the countable, but infinite, set.
What I don't understand is why the new guests can't be moved straight to $n+1$, where $n$ is the final room. Or have the current guests at $n$ move to $n+1$ and the new guests move to $n$.
Is it due to $n$ being inaccessible directly, so that you need to build to it iteratively? Is it a genuine requirement?
If it is, then why not build a function $f$ such that:
$$f : X \to X', f(n) = \left\{ \begin{array} {ll} n, \ \text{if room} \ n \ne 0 \\ \text{new family, otherwise} \end{array} \right.$$ (terminating after the second condition is met, and iterating over $X$ in order beforehand)?
You're assuming that Hilbert's hotel has an empty "$(n+1)$th" room. Note, however, that the entire hotel is full. So, there are no empty rooms.