The rooms are all full by definition. You’re literally telling him to walk until he dies. Just kick him out at that point, hell shooting him on the spot would be more merciful.
But the weird thing about infinite sets is they don’t process over time as we perceive it. Things happen all at once. There’s a classic situation where an infinite tub is filled with an infinite amount of numbered balls, and for every number that goes in, it’s square root is taken out. So when 1 goes in, 1 goes out, when 2 goes in nothing happens, same with 3, when 4 goes in, 2 gets taken out, when 9 goes in, 3 gets taken out.
How many balls are in it at the end?
Intuitively it seems there must be infinite balls, as balls are being taken out at a slower rate than they’re entering. But the actual answer is 0. Because the process happens all at once. So the question becomes, which balls get removed in this process? Well, the numbers that can be squared. Which is every number. If every number can be squared, every number gets removed. So if this infinite process were to play out, there would be no balls left in it.
But if we were to try to physically do this, it’s impossible for it to actually play out. Infinity isn’t a number at the end of a line, it’s the concept of an entire unbound set. That’s why things like the hotel are good to try to explain and visualise the concept, but break down if you try to imagine them as real world places that follow time and physics.
Rainbow gets you to 7, which I can do without thinking: red, orange, yellow, green, blue, indigo, violet. 2 biggest non rainbow colours: pink, brown. Then you just need to name any of the more unique colours: teal, cyan. That’s where I got to before I had to actually stop and think. You could probably add another to the rainbow by listing purple as distinct from violet. You should probably avoid black and white in case the person is being pedantic.