1. Equivalence of two sets

Please I need help proving the following question:
Given an infinite set A and a countable set B, prove that $\displaystyle {A}\cup{B}\sim{A}$.
I already proved the case when $\displaystyle {A}\cap{B}=\emptyset$ and when $\displaystyle {B}\subset{A}$.
I need help proving what happens when $\displaystyle {B}\nsubseteq{A}$ and $\displaystyle {A}\cap{B}\neq\emptyset$
Well, $\displaystyle B=B_1\cup B_2$ for some $\displaystyle B_1, B_2$ such that $\displaystyle B_1\subseteq A$ and $\displaystyle B_2\cap A=\emptyset$. The sets $\displaystyle B_1$ and $\displaystyle B_2$ are countable. Therefore, $\displaystyle A\cup B=A\cup B_1\cup B_2=A\cup B_2\sim A$ by what you have already proved.