Suppose we have one variation of Hilbert's hotel problem:
Suppose we have infinitely many hotels, each with infinitely many rooms, and we want to close all other hotels except one, and we need to accommodate all guests from the closed hotels to the single open one. We are tasked to explain that it is possible to do so, and we need to describe how we would accommodate a guest $(n,m)$, who currently resides in the $n$th room of the $m$th hotel.
One explanation would be if we have a matrix-like structure whose rows would denote the hotels, and whose columns would denote rooms, so that each cell is a $(h,r)$ pair, where $h$ is hotel and $r$ is room. Next, we sort all pairs $(h,r)$ by the sum of $h+r$, of course ascending. The list would go as follows: $(1,1), (1,2), (2,1), (1,3), (2,2)...$
Next, we simply assign all the natural numbers to each pair, so that $1 \rightarrow (1,1)$, $2 \rightarrow (1,2)$ etc, so that each guest from hotel $h$ and room $r$ will be assigned to room number $n$ which corresponds to the pair.
My question is: What is the point of sorting the pairs? It may be a naive question, and admittedly I can't find another way to do it, but it's evident that this matrix is an infinite one so why couldn't we just simply said that it is possible?