Karl Popper maintained that empirical sciences should be based on the principle of falsifiability rather than verifiability
The common-sense attitude, which is that of most people when dealing which everyday situations, but also that of scientists, is to start from factual assumptions, i.e., observations. For example, one would normally start from the assumption that the sun is the brightest object visible in the sky during the day. From this, humans typically conceive a theory, what we could call a theoretical assumption. For example, that the sun turns around the earth. Why not the other way around? Well, we observe the sky for several days and we invariably conclude that the sun certainly appears to be turning around the earth. Thus, adopting assumptions already involves verification. It is definitely true that the sun is the brightest thing visible in the day sky, and that it seems to turn around the earth.
From our assumptions, we may derive all sorts of logical consequences. If we do, we can use them to falsify our theory if we can verify that at least one consequence is false. Doing this again involves verification.
Why would we choose to falsify the theoretical assumption rather than our factual assumption? Well, facts are often just more certain than the theory. More fundamentally, all theories have to be based on some facts. We cannot think that the sun turns around the earth unless we already have a notion of what the sun and the earth are.
So, I'm afraid that verification comes first.
Yet, verification may be mistaken, which is why falsification comes in handy. However, it would be wrong to think that falsification itself cannot turn out to be wrong, too. We could perhaps, for example, be happy to have falsified the theory that Jim was the murderer, but later falsify our falsification by finding some more compelling evidence that he really was the murderer. Our falsifications are usually themselves complexes of facts and theory, and they can often for this reason be themselves falsifiable, if not necessarily falsified.
How falsifiable are assumptions based on probability
All our assumptions about the physical world are based on observations and are therefore fundamentally probabilistic and falsifiable. Which is also why we can doubt even that which we think is most certainly true.
when x tends to infinity; how can one make infinite observations?
We don't know how complex is reality, but if it was infinitely complex, no amount of cogitation of a human brain would possibly get to a true science. No amount of verification and falsification would ever be able to help us produce true science. For one thing, conception would take longer and longer as we proceed to complexify our theories to match the complexity of the world.
Still, we usually assume that reality is not infinitely complex, which means that successive models based on observation, and a logical analysis involving verification and possibly falsification, should get science closer to be true of the real world.
However, even if reality is not infinitely complex, which seems at least more plausible, we still don't know how complex it is, so that while science may get closer to be true, there might not be enough time left in the universe for us to produce true science.
If we further assume that reality is reasonably simple, which is a completely unwarranted assumption, we can still hope to produce true science, and we will do it using verification and falsification. The more complex the reality, the longer it will take to produce a true science of it, but part of current science is very plausibly already true even if we don't know that it is.
We also cannot tell in advance what kind of model will be true of reality. Maybe the basic concepts of General Relativity, though apparently verified, are in fact completely wrong. Logical reasoning at least seems to guarantee certainty that the consequences of a theory fit our basic assumptions, but we conceive our theory to fit our assumptions, so we cannot then use the theory to verify the assumptions, and so our assumptions will always be possibly false, even when in fact true.
Infinity in the size of the universe may or may not be a problem. Simply, if the universe has a size which is infinite, we won't be able to either verify or falsify that our theory applies everywhere in it. We wouldn't even be able to verify our basic assumption that the universe is consistent. And, there is no good reason to assume that the universe is finite in its dimension.
Still, what really matters is to be able to use science to help humanity survive, and any improvement in our scientific theories will help in this respect, even if we don't ever produce a completely true science because the universe, maybe, is either infinitely complex or infinitely big.
Metaphysics may be fun, but it won't help humanity survive... unless reality is really very, very different from what very nearly all humans believe that it is.