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Nitrogen molecules are larger than oxygen, so they don't seep past the rubber and lose air nearly as quickly. Also, nitrogen doesn't expand or contract like oxygen, so tires pressures stay more consistent.
You give two reasons: The first one is easily debunked, in my humble opinion...
If a tire lets O2 out, but keeps nitrogen in. It's like a sieve. Or a filter.
So, after three or four fills of regular air, it should have a MUCH higher percentage of nitrogen to oxygen, because more oxygen is leaking out than nitrogen.
Which would make paying extra for nitrogen fills, silly. You'll end up with nitrogen after a few checks of the tire pressure anyway.
If this worked, one could just keep a few old tires inflated in the corner of the shop, wait until they deflated 20% (oxygen is roughly 20% of the volume of air -- let's not go into humid vs. dry...) and then you'd have free nitrogen to put in your other tires!
I've been using regular old "free" air since I could drive. Holes that are large enough to let even nitrogen molecules out are the only times I have been let down, literally.