This is yet another cool variation on the classic infinite hats/boxes/prisoners riddle. If you solved one of the other variations (such as this one or this one) this shouldn’t be too hard.

You enter a room with infinitely many boxes. Each box contains a single natural number inside. You are allowed to open as many boxes as you want and look at the numbers inside, as long as you keep at least one box closed. At some point, you must guess the number inside of a box you haven’t yet opened. How can you guarantee to guess correctly with a probability of at least 99%?