If you look at your AA flight reservation, there's a box you can click that says "Get ready to fly". If you click on that, it lists your destination's requirements. That's where, if you keep reading and scrolling down, the statement about the required funds is located. It doesn't pop up on its own (at least not on my phone). Regardless, you don't have to prove anything about your financial situation before you fly. It's just the airline trying to prepare you for what might be required when you arrive. BTW, it does state that there is a mandatory electronic immigration form.
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