In the reals, although 'Infinity' is not reachable, is it approachable?

For example, is 20 closer to ∞ than 0?... I'm thinking no. The way I'm thinking about it is I'm considering an 'infinite hotel.' We have a Lobby, Rm 1, Rm 2, Rm 3, Rm 4, and so on. A start, but no end. Now, in this hotel, every room is an integer #. For example, there is no room #∞. The thing is, what if I ask "the first 20 guests to leave." Now, rooms (1 - 20) are empty. Now, I ask all the other guests to move to the left 20 rooms. So,.. guest in Rm #21 is now in Rm #1,... guest in Rm #22 is now in Rm #2,... guest in Rm #23 is now in Rm #3, and so on. The thing is, every room occupied prior to the guests leaving is still occupied now. If 20 were closer to ∞ than 0, there would be less rooms filled.