Finding a finite integer in infinite space An adversary selects an integer k from the set of non-negative integers.
Does any algorithm exist that, using only tests for equality or inequality (<, =, >), is guaranteed to find k in finite time?

At step $n$ test the non-negative integer $n-1$ for equality; you are guaranteed to find $k$ on step $k+1$, so the algorithm is guaranteed to terminate after a finite number of steps, though the possible stopping times are unbounded. What is not possible is to put a fixed limit on the search time beforehand: no matter what algorithm you use, for any $n$ there will be values of $k$ that cannot be found within $n$ steps.
