I need to write a proof that, given A is a finite set and B is an infinite set, show that A is equinumerous to a subset of B. I understand to show this I need to show a bijection.
I can get the injectivity easily. What I don't understand is how A is how to show A surjective to a subset of B. By the definition of surjection that for each y in B there must exists an x in A such that f(x) = y. Can I just state that is MUST be a subset or else it wouldn't be surjective and therefore A would fail to be equinumerious to a subset of B?
Can I even apply those definitions because they involve a function when the question doesn't mention functions at all?
You will need to define a function from $A$ to $B$ and then show it is a bijection.
I don't know how technical of a proof they are looking for, but it seems trivial to just say that if $A$ is finite (say, $A = \{ a_1, a_2, ..., a_n \}$ for some number $n$), and $B$ is infinite, then obviously there will be a subset $B'$ of $B$ of size $n$, and so you can say $B' = \{ b_1, b_2, ..., b_n \}$. And then you can define $f(a_i) = b_i$ for all $1 \le i \le n$, and show that this $f$ is a bijection.