Let X be a metric space. A self-map on X is said to be a pseudo-contraction if d(P(x), (y)) d(x, y) holds for all distinct x, y X Show that if (m (x)) has a convergent subsequence, then d(x*, (x)) = d($(x*), (x)) for some x* X We can use the property: if DEXX is a pseudo-contraction, then (d(m+1(a), (x))) is a decreasing sequence, and hence converges.