In this example we construct a sequence {f} on [0, 1] that converges to zero in the mean, but for which {f(x)} does not converge for any x = [0, 1]. This se- quence is constructed as follows: For each nEN, write n = 2k +j where k = 0, 1, 2,..., and 0 j < 2. For example, 1 = 2 + 0,2 = 2+ 0,3 = 2 + 1, etc. Define f, on [0, 1] by f(x): = j+1 2k sxs (0, otherwise.