Convergence of the median filterings of sequences

Suppose that x = {x(n)}(n is an element of z) is a sequence of real numbers . For each p is an element of N, x((p)) = {x((p))(n)}(n is an element of z) is the resulting sequence of x through p times median filterings with wind ow 2k + 1. It is proved that when p-->infinity, both x((2p)) and x((2p-1)) are convergent. Thus the problem of convergence of the median filters of in finite-length sequences is completely solved.