A note on tameness of families having bounded variation

We show that for arbitrary linearly ordered set (X,≤) any bounded family of (not necessarily, continuous) real valued functions on X with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space (Y,d) the compact space BVr(X,Y) of all functions X→Y with variation ≤r is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space M+(X,Y) of all order preserving maps X→Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.