 Each "g""n" is continuous, and so each "E""n" is open (because each "E""n" is the preimage of an open set under "g""n", a nonnegative continuous function).Since { "f""n" } is monotonically increasing, { "g""n" } is monotonically decreasing, it follows that the sequence "E""n" is ascending.Since "f""n" converges pointwise to "f", it follows that the collection { "E""n" } is an open cover of "X".