I am a beginner on martingales and on my notes I have that a martingale bounded in $L^1$ has a finite number of upcrossings of an arbitrary interval $[a,b]$.
Why the martingale cannot cross the interval as many times as it wants? I don’t need a formal proof but it would be enough just an intuitive explanation. Thanks in advance
The answer above is circular to me (since the proof of martingale convergence that I know goes through the upcrossing inequality), so I will try to give a different answer (I will be paraphrasing Durrett PTE version 5 Theorem 4.2.10 and 4.2.11, so feel free to read those up; I think they are very accessible).
The intuition is the following: since a martingale is a "fair game", you should not be making money betting off of it, and if there were infinitely many crossings, and say you bet 1 every time it crossed, then you could reasonably play this game again and again and make infinite money off of it.
The only snag in this logic is that you might cross it infinitely many times, but then if it could take infinitely long, you still could be in the "fair game" setting. If your martingale is L1 bounded, then this is not going to happen.