First you need to describe what 'deep' is. It is relative for most, for example 'deep' to me is 200 fsw, for some of the CCR divers that I frequently dive with this is 'recreational' since it is within Normoxic limits and 'deep' to them is 460 fsw. OK, back to your question.
Nitrox (EANx) LIMITS your depth, and does not allow you to dive deeper. This has to do with the partial pressure of oxygen (PO2). Max on PO2 that you want to breath for a short time is 1.6. So lets look at the two common EANx mixes that you will learn about in your class.
EANx32 (32% oxygen, 67% nitrogen, 1% other but for argument sake, 68% nitrogen). This will give you a maximum operating depth (MOD) of 130 fsw (PO2, 1.6)
EANx36 (36% oxygen, 64% nitrogen) gives you a MOD of 110 fsw (PO2 1.6)
As you can see the higher the oxygen the lower the MOD. The benefit you get from higher O2 is less Nitrogen, thus less narcosis and less formation of bubbles. The less Nitrogen the more the gas acts like you are more shallow, termed EAD, equivalent air depth. You want your EAD to be 130 or less, I like my gas to be 100 or less.
How 'deep' do you consistently dive? Do you get narked at this depth? Have you felt what being narked feels like? If you really want to go 'deep' you can now see why you can't just replace the nitrogen with oxygen because it would cause an O2 hit. So, this is why you need to replace it with something else, Helium. Once you decide to go deep, take a 'recreational trimix class' IANTD has one. This keeps you to a depth of 140 fsw but allows you the benefit of a low EAD and a clearer head while at depth.
Now to your BO bottle (40 cuft). If you are staying above 130ft, fill it with EANx32. Even if your computer is set to 'Air' you are getting the benefits of higher O2 and less saturation of tissues as you start your ascent. (I would not use EANx32 for the bottom gas due to building up your OTU, but the BO is fine to switch to at depth and start your ascent).