Fitting new guests into a full infinite hotel

A hotel has countably many rooms numbered $1,2,3,\dots$, and every room is occupied. Explain how to make space in each case:

(a) one extra guest arrives;

(b) a countably infinite line of new guests arrives;

(c) countably many coaches arrive, each with countably many passengers.

(a) Move each guest from room $n$ to $n+1$, freeing room 1. (b) Move each guest from room $n$ to $2n$, freeing all odd rooms for the newcomers. (c) Move current guests to $2n$; then place passenger $j$ of coach $i$ in room $3^{i}5^{j}$ (or use any bijection $\mathbb N\times\mathbb N\to\mathbb N$), which is odd and never repeats by unique factorisation.