Fun fact: it does fall down ! Everything falls down due to gravity. Think of the atmosphere as another ocean, the fluid dynamics you can see in water apply to all the gases around us too.
What we consider air and oxygen are floating in the gas soup at ground level and they have specific densities which causes them to separate... i.e. when you climb a mountain oxygen levels go DOWN!
eh, not really. mountains aren't high up enough for this to matter, there's less oxygen at mountaintops because there's less air there in general (less air pressure). you need to go all the way to the top of atmosphere for this effect to matter
for example, if you release helium or hydrogen, it will eventually float up to the top of atmosphere and because it's so light, it moves fast and at some point it will reach escape velocity and drift away into space
When they said oxygen levels go down, they presumably meant that the partial pressure goes down. That’s what people probably think of anyway since that necessitates supplemental oxygen (if you are high enough).
The supplemental oxygen is needed because the total pressure is down, not because of gas separation. Humans need some number of moles of oxygen per minute, and at the top of Everest they just cannot ingest that number fast enough because the total pressure is so low. The percentage of oxygen in the total gas mixture is also lower on Everest than the 21% at sea level due to the aforementioned specific density separation, but that effect is much smaller and insignificant compared to the total pressure shortfall. This is why the grandparent comment objected to mentioning it - this is an interesting fact on its own, but irrelevant to the question of supplemental oxygen on Everest.
Yeah, on rereading I see that I originally missed the specific mention of densities causing them to separate (instead of just “less oxygen”), so I now agree with the objection.