Prove the following implication by induction on $m$:
If there exists an injection $N_m \rightarrow N_n$, then $m \le n$.
$N_n$ and $N_m$ are sets with $n$ and $m$ elements, respectively.
I know how to do proofs by induction, I'm just struggling with the inductive step. So far I have the predicate P(n) = If there exists an injection $N_m \rightarrow N_n$, then $m \le n$.
I proved the base case "P(1) = If there exists an injection $N_1 \rightarrow N_n$, then $1 \le n$" sloppily by saying no element in $N_n$ could be assigned to more than one element of $N_1$ because there is only a single element in $N_1$. There must at least one element in $N_n$ in order for it to be a function, so $1 \le n$.
So I know the inductive hypothesis is P(k) = If there exists an injection $N_k \rightarrow N_n$, then $k \le n$. Now I need to prove P(k+1) = If there exists an injection $N_{k+1} \rightarrow N_n$, then $k+1 \le n$, but I'm clueless about how to do this step.