Which is the simplest example of a subset of R that is not a Borel set?
Ps: I asked my probability professor how I can see if a subset of R is a Borel set or not? He answered that in the case we will use the set are always Borel set and in fact he said that all subset of R are Borel set. So at this point I was very interested: why I can't imagine a subset of R that is not a Borel set? So I asked him an example of a subset of R and he answered that he will not do that because it will only confuse me and because is not important. So I'm asking here!
Ps2: I don't have any background of topology, geometry and so on. I know analysis (I-II) and linear algebra (I-II) so keep it simple please.