This is a common misconception, in standard models of big-bang cosmology the universe was always infinite. It was hotter and denser earlier, and is cooler and less dense later but always infinite.
A decent mental model is to think about an infinitely long ruler or measuring tape, where I have positions marked every (e.g.) 1 meter interval. You can imagine stretching or shrinking the ruler which would move the marks further or closer to each other respectively.
If you think about a "cosmology" for this ruler where I keep stretching it further and further forever then the points I have marked will ger further and further away, but the ruler itself is always infinite.
That's not how physics works. We can't prove anything. All we do is come up with worse or better models, and the best model anyone has come up with so far has the universe being infinite. Finite universe models have some problems, since the global curvature of the universe is (as best we can measure) zero the universe is either flat or pretending to be flat. Finite volume flat spaces can be characterised and they're a bit weird (1).
The best we can say is that if the universe is not infinite it appears to be doing a very good job of pretending to be infinite.
(1) for a good idea of what "weird" means here imagine a 2d 1km by 1km square with geometry like Pac man lives in (toroidal), so if you go off the top of the square you appear at the bottom, and if you head off east you appear on the west. Now start at the middle and leave a rock at your current position. Head due east until you cycle round and hit your rock again. You'll have walked 1km (500m to the east edge and 500m from the west edge back to the middle). Now do the same experiment but walk north-east. You'll hit your rock after sqrt(2) km of walking (1/sqrt(2) takes you to the north east corner and the same to get back to your rock).
In other worlds the pac-man torus space is not isotropic some angles are special and more important than others. Essentially the same thing happens in other finite flat geometries you can invent.
I don't say it has to be isotropic, but observationally it looks very close to isotropic. There are quite clear signals we'd expect to see in the cosmic microwave background if the universe was anisotropic and they aren't there:
This study is (as far as I'm aware) the state of the art on this topic, and based on the CMB observations they use it appears the universe is incredibly close to isotropic. Note that because it is based on CMB data this sort of study is sensitive to what shape the universe was a long time ago when, if it is finite, it was a lot smaller.
A decent mental model is to think about an infinitely long ruler or measuring tape, where I have positions marked every (e.g.) 1 meter interval. You can imagine stretching or shrinking the ruler which would move the marks further or closer to each other respectively.
If you think about a "cosmology" for this ruler where I keep stretching it further and further forever then the points I have marked will ger further and further away, but the ruler itself is always infinite.