I am reading through a proof that the Vitali set is not measurable. I have gotten to a line that says
$$[0,1] \subset \cup_{q \in [-1,1] \cap Q} (A+q)$$
and I don't see why this is true. Clearly, the set of $A + q$ sets are disjoint, but I don't see why the above line holds. If $\eta \in [0,1]$, $\eta -a = q$, for some $q \in Q$ and some $a \in A$, but I guess I do not see why this $q \in [-1,1]$.
If $a \in A \subset [0,1]$ and $q = \eta - a$ with $0 \le \eta \le 1$, then $-1 \le \eta - a \le 1$.