# Why is $BPP^{NP}$ in the polynomial hierarchy?

Why is $$BPP^{NP}$$ in the polynomial hierarchy? I know that $$BPP$$ is contained in $$NP^{NP}$$, so $$BPP$$ is inside $$PH$$. Should I just simply reuse the proof of $$BPP\subset NP^{NP}$$, feed in each machine an oracle solving 3SAT problem, and show that $$BPP^{NP}\subset NP^{NP^{NP}}$$? Is it because normally, the proof in complexity theory can be easily relativized by an oracle?

• Have you tried following your suggested proof technique? – Yuval Filmus May 8 '19 at 11:49
• You already got advice on how to prove it: cstheory.stackexchange.com/a/43843/5038. domotorp suggested exactly this. Now it sounds to me like you're coming here to ask "should I do what I was advised to do?". Well, I would think a reasonable answer is "yes, give it a try and see what happens, probably they had a reason to suggest that; and then if you run into some specific step where you get stuck, ask about that step". Have I misunderstood the situation? – D.W. May 8 '19 at 16:47