Since particles not at absolute zero have finite energy, this point is non-trivial and proving this point was actually the focus. (Tangentially, I think this has been a common and strong intuition, and suggested by extrapolation from empirical evidence, that you could asymptotically approach but not reach absolute zero because it would be progressively harder as you got closer. But there's a difference between that and a mathematical proof grounded in assumptions that are well-accepted.)
Yes, if you have this as a known truth, you're basically done.
That's the hard bit. The infinity could simply be an artefact of your description. You could for instance describe direction on a map as 'miles east / miles north' and conclude that it's impossible to move due east or west but that's clearly not the case.
A more sophisticated example would be the event horizon of a black hole, where several quantities seem to become infinite, but it turned out this wasn't the case with the right coordinate system, meaning that it's likely all too possible to cross the event horizon.
Like a really basic proof is:
- It takes an infinite amount of time and resources to remove energy to reach absolute zero
- You can't reach infinity
- Therefore reaching absolute zero is impossible.
What deeper knowledge am I missing?