It depends on your definition of proof.
Proof in everyday usage is just some sufficiently convincing evidence, and what is sufficient depends on the person who wants proof of something. For most people there are many things that can be proven beyond reasonable doubt. For example, the earth is not flat and the planets in our solar system orbit the sun and not the earth. For another example, you were not constructed just one millisecond before the present moment and implanted with memories that fool you into thinking that you have been alive for a long time. (Wait a second and consider this possibility again! Maybe just before now you were constructed and implanted with the memory that you were reading about this...)
However, if you take the most stringent possible definition of proof, an argument that can under no circumstance be incorrect, then there are not many things that can be proven. One example is the fact that something exists. No matter what, this present fact cannot be denied, even if we cannot absolutely prove much about what kind of thing exists.
Even mathematical proof is not absolute, but is just a sequence of logical deductions, each one based on axioms or prior deductions and derived by some inference rule. A mathematically proven statement would be absolutely correct if all the axioms and inference rules used in the proof are first accepted as absolutely correct. That is the whole purpose of creating formal systems, so that anyone who accepts all the axioms and inference rules will have no choice but to accept all the statements proven within the formal system. The fact is that without any inference rule, there can be no formal proof at all. And the less axioms are accepted, the less can be proven. One must start with at least one initial assumption otherwise one gets nothing. Incidentally, essentially all formal systems include the assumption that something exists in the universe of discourse, because otherwise there is nothing to talk about and no reason to talk at all!
That said, there is an underlying explanation as to why some people as you describe do not believe in truth and yet believe that there is always a chance for anything to happen. First one must make a distinction between people who believe that there is an absolute truth that is inaccessible and people who believe there is not even absolute truth. The former is reasonable, but the latter is not, because it is inconsistent. The very concept that there is no absolute truth is itself an absolute concept, and so if it were true then by itself it cannot be absolutely true, which is self-contradictory.
Note that one must also avoid the fallacy that anything that is possible must happen, since it is equally self-contradictory. Specifically, it is possible that not everything possible does happen, which would contradict that fallacy if it were true. Therefore it cannot be true.
Therefore it does not make sense for people to claim that there is a chance for everything, because given any statement that is precise and well-defined in its context, either it is true or it is false. The fact that we are normally rather imprecise, due partly to our use of natural language, does not mean that one is justified to disregard everything that anyone else says. Rather, one ought to make a reasonable attempt to figure out the underlying precise statement before attempting to judge whether it is true or false.
Such claims of the complete subjectivity of truth are often because they either do not want to care about the truth or falsity of the statement under consideration, since it takes a lot of time and effort to investigate truth, or they have ulterior motives in affirming certain so-called possibilities.
The clearest examples of instances of people evading the truth are precisely the ones concerning important truths, such as moral atrocities. Usually, the perpetrators and others who have vested interests in covering the truth up will continually and vehemently deny, and sadly in some cases the truth is gradually lost over time.