1. The problem statement, all variables and given/known data Show that for a set S, there exists an injective function [itex]\Phi[/itex] : N [itex]\rightarrow[/itex] S if and only if there exists an injective, but non-surjective function f : S [itex]\rightarrow[/itex] S. (Sets S satisfying this condition are called innite sets.) 2. Relevant equations 3. The attempt at a solution Since this is a if and only if (biconditional) statement. I can prove this statement if i can prove the two conditional statements: i) If [itex]\Phi: N \rightarrow S[/itex] is injective then f: S [itex]\rightarrow[/itex] S is injective but not surjective ii)If f: S [itex]\rightarrow[/itex] S is injective but not surjective then [itex]\Phi: N \rightarrow S[/itex] is injective. I realize that this is the step that i should take, but i just don't know how to prove these two statements.. Any help?