Consider that we define an "infinite" set as one which contains some proper subset that is in bijection with itself. Now suppose that $B$ is a set and $A \subset B$ is an infinite proper subset of $B$. Can we prove using just that definition and elementary facts about functions that $B$ must also be infinite?
You are assuming $A$ to be infinite according to your definition, right?
Let $C$ be a proper subset, and $f : A \to C$ a bijection.
Consider the proper subset of $B$ given by $D = C \cup (B \setminus A)$.
Now consider the bijection $g : B \to D$ which is the identity on $B \setminus A$, and $f$ when restricted to $A$.
If you are assuming that $A$ is infinite, then yes, we can. In that case, there is a bijective function $f:A\to C$ with $C\subsetneq A$. Let $g$ be the identity function on $B\setminus A$. Then $f\cup g$ is a bijection between $B$ and a proper subset of itself, namely $(B\setminus A)\cup C=B\setminus (A\setminus C).$