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?
 A: If player A picks a number from a countable set, then player B will definitely get there by guessing systematically. Suppose we're playing it, and the number I pick is 9000000000000: you start guessing at 1, then 2, 3, etc., you'll eventually get to 9000000000000. So the game will last a finite amount of time.
If player A picks a number from an uncountably infinite set, then player B will (most likely) never get there by guessing systematically. Suppose we're playing it, and the number I pick is 2. If you start guessing 1, 0.1, 0.01, 0.001, 0.0001, etc. then you will never get to 2. The chance of correctly guessing my number is arbitrarily small, though I hesitate to say that you'll never guess my number (because we could play the game, I could pick a given number, and you could simply guess it). So in this case the game may last forever.
What this game really gets at is that there's a bijection from the natural numbers to any countably finite set, but there's no bijection from the naturals to the reals.
A: 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)
