Prop: If the set $A$ is infinite then there exists an injective function $z:N→A$. $z$ is defined inductively, for the base case, explain how it is plausible to define$ z(1)$. Now suppose that $z(1),z(2),…,z(n)$ where no two values are equal. Prove that it is possible to define $z(n+1)$ where its value is not equal to any previous value.
I now understand that this has a never-ending injective function that can be created, however, I do not understand how to actually write out the proof. I have written induction proofs before but not seeing an equation or sigma notation to follow I feel quite lost setting up a base case or inductive step showing $z(n + 1)$.
A written proof with explanation would be much appreciated!
Link to similar question: Proofs involving functions and induction