Helly's theorem and piecewise constant function

muzialis

Hello there,

I found in a book a proof of the fact the space of piecewise constant functions with a finite number of jumps is compact, hence a convergent subsequence can be extracted from any bounded sequence.
I am wondering , is that not a direct consequence of Helly's Selection Theorem?
Indeed, seems to me any piecewise constant function with a finite number of discontinuities belongs to BV locally, which is the requisiste in the aforementioned theorem. Am I worng?

Many thanks