Is the universe infinite?
We don't know, we can't know. And we never will know. But there are some solid reason to think it is, and some solid counter arguments.
One argument is based on the fact that we look out at things far away and notice they are all moving away from us. We argue that it is unlikely we happen to be at a unique place in the universe so we ask ourselves how the universe would look if that same observation came out the same way in many places in the universe, so if it were a generic real estate feature.
When we restrict ourselves to such models we cab design models of how our universe looks and then on large scales it looks pretty uniform, sure there is a planet right under us and a star pretty nearby but when you look at large enough regions it looms pretty similar here there and everywhere (those this is still a conclusion from the model, we don't actually see what it looks like elsewhere).
When we make models about how universes that are super uniform look like we find three major types, ones that are positively curved, ones that are negatively curved, and some that are flat. And then we can try to measure the large scale curvature and we get a number that could be zero.
Every measurement has error so either it is zero or else it is smaller than our error. It is possible to have a flat and finite universe, but is does make our motion rather special so some people reject it (of course over time a geodesic motion has its peculiar velocity become closer and closer to the Hubble flow plus we are made out of neighbors so of course we aren't moving terribly different than them if they are moving together). So I admit I fail to see how this is evidence for an infinite universe, but I understand this global large scale flatness is the essential reason many scientists think the universe is infinite.
But there is a problem with this evidence. When we look very far away we look at earlier times (it takes time for light to get to us) and things that are too far away seem too related. So people postulate that maybe the universe went through a growth spurt so those places used to be closer together.
However this growth spurt (called inflation) causes the large scale curvature to change over time and to become numerically close to zero even if it is actually positive or negative. So that might be why it is so small now. If the curvature were positive or negative no one would seize that as evidence that the universe is infinite.
So there are people that think the universe is infinite, they'll cite the "we don't live in a special place" principle (which is great) and the possibly zero curvature (which might be explained by inflation) and then there is some hand waving where they conclude the universe is infinite.
But we can't test that anyway, so few people bother to argue. The part that could affect us now is finite and the last we can affect is finite. So whether it is infinite or finite doesn't affect our predictions so it is a bit out of scope for science.
But its not like it is better if you claim the universe is finite. After all even if you walked a huge distance and came to a place that looked just like your house including all the views from your house did you walk around an finite universe or does the universe have a regularity where it tends to looks similar if you go a certain distance. If it was really finite and you walked around again you'd get back to your starting point but if it is just likely to be regular then any of those trips might end up someplace that looks different.
It's like some ants going down a long street of identically built identically landscaped identically furnished row houses. Near the middle of the street the universe might look finite and rather small. But as they get near the edge they can start to see the view looks different. As they get to the end of the street it is very different looking.
So even if you saw evidence that was 100% exactly like what you'd expect for a finite universe the universe still might be bigger or even infinite.
So no evidence is good enough to show it is infinite and no evidence is good enough to show it is finite.
But making some choice for your model might make it easier or harder. But don't get too excited either way. You only make finite measurements, and from them you will never know whether the universe is infinite or finite.
You can definitely rule out it being tiny and we've done that. You can even rule out is being medium sized.
So the universe is big, and that's good enough.