Nobody has proof, and it's probably unknowable. It could be that we're a discrete system embedded in a continuous system. Or it could be we're in a seemingly continuous system which is actually embedded in a high precision discrete system.
That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length. Because of uncertainty principle, we can't measure distances less than that. In a similar way, time also appears to be discretized, along with energy and mass.
The fact that we don't know what's going on within the Planck length doesn't demonstrate that spacetime is discretized; I don't even think it strongly indicates it, unless you're already predisposed to think that way. Anyway, that seems to have no bearing on whether the universe is infinite.
That being said, nature has given us very strong hints at discretization. For example: https://en.m.wikipedia.org/wiki/Planck_length. Because of uncertainty principle, we can't measure distances less than that. In a similar way, time also appears to be discretized, along with energy and mass.