Sequential properness for a self-adjoint operator

For a bounded self-adjoint operator $L$, every bounded sequence whose images converge has a norm-convergent subsequence exactly when its kernel is finite-dimensional and its range is closed, or equivalently $0\notin {\Sigma }_{\mathrm{e}}(L)$ for the essential spectrum of a bounded self-adjoint operator. Split the sequence into kernel and kernel complement: finite dimensionality gives a subsequence on the first part and the closed-range bound on the kernel complement makes the second part a Cauchy sequence. Infinite kernel or approximate null unit vectors in its complement obstruct the property.