Assuming (x_n)_{n Î N} as an orthonormal basis for a separable infinite dimensional Hilbert space H, say l². Denote the linear span of {x₁, x₂, ¼, x_n} by Y_n. Let P_n be the projection onto the closed subspace Y_n. If m < n, then Y_m Ì Y_n and so 0 < P_m < P_n. Moreover P_n - P_m is a projection and so ||P_n - P_m || = 1 whenever n ¹ m. Therefore (P_n) is an increasing sequence of self-adjoint operators which is not even a Cauchy sequence in uniform topology on B(H).