Problem
COM-B2-M04-P001 A Chain of Larger Numbers
#1
★★☆☆☆ Level 2 of 5
In a finite nonempty set of positive numbers, to each number \(x\) there is assigned a number \(f(x)\) from the same set such that \(f(x)>x\). Prove that this is impossible.
Choose the largest number in the set.
Let \(M\) be the largest element of the set. By the condition, \(f(M)\) belongs to the same set and \(f(M)>M\). This contradicts the maximality of \(M\). Therefore no such assignment exists.
A basic problem for understanding finiteness and choosing a maximum.