So currently I have a problem on the following question: Let $A$ be an infinite set. Show that there is a countable subset $B\subseteq A$ such that $|A|=|A-B|$ as asked here: Subset of an infinite set with same cardinality.
The construction of a set $B$ seems pretty easy (One thing I have noticed is that $B \neq A$). But showing that there is bijective function with $A-B \rightarrow A$ is where I am currently stuck. We cannot use a slightly altered version of $id_A$ as there is no good way to "hit" the missing elements in $A$ I guess.
So my question is: How can I show that there is such a bijective function or how am I able to construct it?
As this is homework I would appreciate hints over full solutions for now.