@Mike @TedShifrin
exact words from my prof when I emailed about the problem....

In the next sentence you talk about $A_\infty$ and
$B_\infty$ being injective and surjective, That does not make sense, $A_\infty$ and $B_\infty$ are sets,
not maps. So what do you need.

1.) Show that $f_\infty$ is defined, i.e., if $a \in A_\infty,$ then $ f(a) \in B_\infty$.
2.) Show that f_\infty is injective, clear because f_\infty is a restriction of the
injective function f.
3.) Show that $ f_\infty$ is onto, i.e.,$if b \in B_\infty,$ then there is an $a \in A_\infty$

anyway...


1.) Show that$ f_\infty$ is defined, i.e., if$ a \in A_\infty$, then $f(a) \in B_\infty.$
2.) Show that $ f_\infty$ is injective, clear because $ f_\infty$ is a restriction of the
injective function f.
3.) Show that $f_\infty$ is onto, i.e., if $b \in B_\infty$, then there is an $a \in A_\infty$
and $f(a) = b$. So, show that there is an appropriate element in$ A $ that maps to
b, and then show that it even belong to $X_\infty.$

@FernandoMartin those are the three hints I was given and each of them is only 2 lines