I'd like your help with the following question:
Assume we proved that $\mbox{BPP}\subseteq \Pi_2$ -What conclusions can you make?
BPP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1/3 for all instances,
$\Pi_2$ is the class of all languages $L$ such that there's a polynomial algorithm $M$ and a polynom $p$ so that $\forall x.x\in L\Leftrightarrow \forall u\in \{ 0,1 \}^*.\exists v \in \{ 0,1 \}^*.M(x,u,v)=1$.
We already know that $\mbox{BPP}\subseteq \Sigma_2$, so $\mbox{BPP}\subseteq \Pi_2\cap \Sigma_2$.