I can't think of an easy proof but it is clear that such a sequence of estimators, {V_n}, has the median as a fixed point in the following sense.

Assume V_(n-1) = median(X) for the data stream X_n. Now we have

    E[V_n | V_(n-1)] = V_(n-1) + E[signum(V_(n-1) - X_n)]

but by the definition of the median, the right hand side expectation is 0, so the random process is fixed in expectation.

Proving it'll converge to that value would be harder, though. It'll probably depend on the data being much larger than the step size (which is 1).