A bounded increasing sequence of self-adjoint operators on a Hilbert space which is not uniformly convergent.

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Assuming (ξ_{n})_{n∈N} as an orthonormal basis for a separable infinite dimensional Hilbert space H, say l^{2}. Denote the linear span of {ξ_{1}, ξ_{2}, ⋯, ξ_{n}} by Y_{n}. Let P_{n} be the projection onto the closed subspace Y_{n}. If m < n, then Y_{m} ⊂ 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 ≠ 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).