Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2010 / B / 4 / v

From $B_k{p}_k=-g_k$ and the matrix 2-norm bound, $\Vert g_k{\Vert }_2\le {\kappa }_b\Vert p_k{\Vert }_2$, so $\Vert p_k{\Vert }_2\ge \Vert g_k{\Vert }_2/{\kappa }_b$. Let $C=2\tau \epsilon (1-\beta )/\gamma >0$. The summand in the supplied decrease bound satisfiesThus, with the fixed positive constant $c=\beta \epsilon \min ({\alpha }_{\mathrm{init}}/{\kappa }_b,C)$,The nonnegative series therefore converges, forcing its terms to zero:The argument uses the given angle condition and upper bound on $\Vert B_k{\Vert }_2$; it does not need an additional uniform lower eigenvalue bound on $B_k$.