How long would it take to guess an arbitrary integer or real number? Let's say two mathematicians play a game. One of them picks an arbitrary element from a countably infinite set (perhaps the integers, as per the title), and the other one guesses what it is. The second player has as many guesses as they need, and after each guess, they are simply told whether they were correct or not.
Would this game never end, or would it last an arbitrarily large but finite amount of time?
What if the first mathematician selects from an uncountably infinite set, such as the real numbers?

In the first case, one constructs a bijection $f$ from the natural numbers into said countable set. Then you guess $f(1),f(2), \dots$ and eventually (after finitely many steps) you guess correctly.
In the second case, with probability 100% you will always guess incorrectly, so usually the game will go on forever. (You can make only countably many guesses anyway)
