I AM LOOKING FOR A HINT, NOT A FULL SOLUTION, TO THE FOLLOWING PROBLEM:
A function $f: [a,b] \to \mathbb{R}$ is called Baire-1 if it is the pointwise limit of a sequence of continuous functions. Prove that a function $f$ is Baire-1 if and only if the sets $\{ x: f(x) > c \}$ and $\{x: f(x) < c \}$ are $F_\sigma$ sets for each real number $c$.
I have no intuition with these concepts. I would very much appreciate a hint.
I would even further appreciate if someone could outline how I can play around with questions like these. I find non-analysis problems much easier to play around with and gain intuition about and I would like to gain a similar ability for analysis.
Small Hint: the sets that you are asked to prove something about are open sets, and open sets can be written as the countable union of closed sets (i.e. $F_{\sigma}$ sets). Each of those sets can then be viewed as part of a sequence of functions that you can take a limit of.
Slightly larger hint: the open interval $(a,b)$ can be written as $\cup_{n\in {\mathbb N}}[a+1/n,b-1/n]$ and we can define functions $f_n$ by $\{x: f_n(x) \geq c+\frac{1}{n} \}$
For general intuition: this is something that tends to come with time. I found I needed to think a lot about these kinds of things before I had a library of tools that let me look at something and know where to start. Hunting for counter-examples tends to help a great deal as it forces you to really understand what you're trying to work with, and finding them can help direct you towards the right route when you get stuck on other problems.