Dini's theorem is commonly seen in real analysis courses (possibly with the requirement that $X$ be a compact metric space if topological spaces are still off in the future), but suppose one wanted to give an example of how much it can fail without the requirement that the pointwise limit of the sequence of functions is continuous. The problem therefore is:
Exhibit a sequence of continuous functions $f_n: [0,1] \to [0,1]$ pointwise monotonically decreasing to a function $f: [0,1] \to [0,1]$ such that the set of points where $f$ is discontinuous has measure $1$.
This could for example be used to demonstrate how poorly behaved the Riemann integral is with respect to pointwise limits since the sequence $(f_n)$ is pretty much as nice as you can possibly get without being uniformly convergent and $f$ is obviously Lebesgue integrable, but it "maximally fails" to satisfy Lebesgue's criterion for Riemann integrability. Additionally, it would demonstrate the existence of comeagre Lebesgue null sets because the set of points where $f$ is continuous is comeagre by the Baire-Osgood theorem (does anyone have a good free online reference for this? (EDIT: I wrote one myself)).
PS: I know of an example myself (and I'll obviously be posting it later if noone posts one); I'm asking in order to have a reference to point to.