We have two non-empty finite sets $A$ and $B$.
I want to show that $| A | \leq | B |$ iff there is an injective map $f$ from $A$ to $B$.
I have done the following:
If $f$ is injective then for each element $b\in B$ there is at most one element $a\in A$ such that $b=f(a)$.
We suppose that $|A|>|B|$. So since at the domain there are more elements that in the image, it follows that at least two elements of the domain must be mapped to the same element of the image. That means that the function is not injective and so we get a contradiction.
Is the proof of the direction $\Leftarrow$ correct?
Could you give me a hint for the direction $\Rightarrow$ ?