I finished my test and there is a question I completely failed but that my teacher did not go over, so I was hoping someone could post a correction of it, so that I understand what I was supposed to do for next time.
Suppose that A is a countable set of real numbers. Show that there exists a real number a such that $(a+A)\cap A=\varnothing$. (Note: By definition, $a+A= \{a+r\mid r\in A\}$.)