Convergent fixed-rank exact sweeps are coordinatewise stationary

Fix one update window. Each entry of its prefix and suffix interface and of its effective Hamiltonian is a finite sum of products of core entries and fixed entries of $H$. Therefore these entries depend continuously on the canonical core arrays. By the assumed componentwise convergence at every stage, the center vectors and effective Hamiltonians at this window converge to limits $x_{\ast }$ and $H_{\mathrm{e}\mathrm{f}\mathrm{f},\ast }$.

For every sweep index $m$ and every unit vector $y$ in this fixed center coefficient space, the exact update at the chosen window gives $x_m^{\ast }{H}_{\mathrm{e}\mathrm{f}\mathrm{f},m}{x}_m\le y^{\ast }{H}_{\mathrm{e}\mathrm{f}\mathrm{f},m}y$ by [F1] and [F3]. Taking the limit as $m\to \mathrm{\infty }$ preserves the inequality because matrix multiplication is continuous, giving $x_{\ast }^{\ast }{H}_{\mathrm{e}\mathrm{f}\mathrm{f},\ast }{x}_{\ast }\le y^{\ast }{H}_{\mathrm{e}\mathrm{f}\mathrm{f},\ast }y$ for every unit $y$. Thus the limiting center minimizes its local Rayleigh quotient and, by [F3], is a lowest eigenvector.

The update-window list is finite. Apply steps 1.1 and 2.1 to each window in that list. Every limiting center is therefore a local Rayleigh minimizer with all other limiting cores fixed, which is exactly coordinatewise stationarity as defined in the Statement. The argument makes no comparison with states outside those local coordinate spaces.