The set S is also bounded above by y/x, so it has a least upper bound α in the real numbers.

Question

Given x = 2 and y = 5, let S be the set of all integers k such that 2k ≤ 5, i.e., S = {0, 1, 2}. Then the upper bound of S is y/x = 5/2, i.e., for any number greater than or equal to 5/2, it is greater than or equal to any element of S. Also, the least upper bound of S is 5/2, i.e., there is no smaller number that is also an upper bound of S. Therefore, α = 5/2. this is best example of The set S is also bounded above by y/x, so it has a least upper bound α in the real numbers.