1. The problem statement, all variables and given/known data Prove that there is no function f:N→N such that for all [itex] n \in N [/itex] , f(n)>f(n+1). There is no infinite decreasing sequence of naturals. 3. The attempt at a solution Lets assume that their is a function f(n)>f(n+1) for all n. This would also imply that f(n) will be mapped to all of N using all n in N. There exists some x in N such that f(x) equals the first natural number. if f(x) gets mapped to the first natural then f(x+1) gets mapped to some natural. This is either the first natural or some larger natural. But this is a contradiction because we assumed that f(n)>f(n+1). so therefore no function exists.