Let $A\subset \mathbb{R}^n$ be a bounded set and $\alpha \in (0,1)$ such that: $\mu^*(A\cap Q)≤\alpha\cdot\mu(Q)$. Show that A is the null set. 
Let $A\subset \mathbb{R}^n$ be a bounded set and $\alpha \in (0,1)$ such that for all cuboids $Q\subset\mathbb{R}^n$: $$\mu^*(A\cap Q)≤\alpha\cdot\mu(Q)$$ Show that A is the null set.

How to begin? What conclusions can I draw from A being bounded?

Let $\{ Q_n \}$ be a countable cover for $A$, then
$\mu^*(A) = \mu^*(\bigcup_n Q_n \cap A) \leq \sum_n \mu^*(Q_n \cap A) \leq \alpha \sum_n \mu(Q_n)$.
Now use the definition of $\mu^*(A) = \inf \left\{ \sum_i \mu(Q_i) \ | \ A \subset \bigcup_i Q_i \right\}$. 
