The question I had in mind in particular was to find a sequence of integrable simple functions that is Cauchy but does not converge for the $L^1$ norm to a simple function- in particular I found a sequence of functions:
$$\bigg\{\frac{1}{n}\lfloor nx \rfloor\bigg\},$$
but I had a lot of difficulty showing that it was Cauchy. I ended up making some kind of hand-wavy argument that allowed me to drop off the absolute value signs in the norm at the expense of adding on a term to the difference. What is the preferred way to go about a proof like this? The triangle inequality doesn't(I don't think) yield a small enough number for each difference in the sequence.