I’d not set out to teach him anything; I’d make accessible mathematics available to him and let him choose what interests him. Enzensberger’s The Number Devil introduces a wide variety of interesting mathematical ideas in a very accessible way. If (or when) his reading is up to it, Martin Gardner’s collections of columns from Scientific American are good.
The main point is that it should be up to him.
There’s all manner of accessible mathematics that might prove to be more interesting or more fun: Fibonacci numbers and their patterns come to mind immediately. Representation in other bases can be fun early on; I especially like binary (as the system that arises naturally when you want the most efficient set of counterweights for an equal-arm balance when the object being weighed and the weights must go in opposite pans) and balanced ternary (as the system that arises naturally when the weights may also be placed in the same pan as the object).
After all, if someone moves to the room further one will never find an empty room.
The first room is clearly empty: there's nobody moving into it.
The only way a person could fail to move into the next room is if they were in the last room. But then there would only be finitely many rooms, so clearly this particular hotel doesn't suffer from this problem.
The main point of this example is to vividly demonstrate one of the ways in which infinite collections differ from finite ones.
(infinite + 1 is impossible, right?)
In this case, we're adding ordinal numbers. And you can add them. The ordinal number describing the hotel rooms is called $\omega$. It is essentially just the sequence of natural numbers.
When you add two ordinal numbers, you essentially just place one after the other. So if we draw a picture of 1:
*
and a picture of $\omega$
then to get $1 + \omega$, we place 1 first, then $\omega$ next:
Looks the same, doesn't it? That's what happens in the hotel. And that's because $1 + \omega = \omega$.
Incidentally, if we add the other way, $\omega + 1$, we get an ordinal number that's bigger than $\omega$. One way to draw it is
The pipe (|) is a decoration to indicate that the .... really refers to an infinite sequence of asterisks (*), and they all are located to the left of the pipe. This is so it's not confused with something like
in which the ... would usually be interpreted as a finite number of asterisks that we were too lazy to write out.
One particularly important thing to note about $\omega + 1$ is that last asterisk doesn't have an immediate predecessor. That's another unusual feature that infinite ordinal numbers (except for $\omega$ have: they can have elements that have infinitely many things before it, but none of them are immediately before it.
Best Answer
To tell someone what infinity ... Take a sheet of $A4$ paper and divide it into two halves. Now take one of the halves and divide it again. Repeat this step indefinitely. Here ask the question 'Will this process finish?'.