= Stationarity from uniformly bounded quasi-Newton matrices
{title2=$\sum_k\|\nabla f(x_k)\|^2<\infty$}
Suppose $f$ is bounded below, $0\prec B_k\preceq MI$ for one fixed $M$, $B_ks_k=-\nabla f(x_k)$, and $x_{k+1}=x_k+\alpha_ks_k$ satisfies the <Armijo condition> with $\alpha_k\geq C>0$. Then
$$
f(x_k)-f(x_{k+1})\geq\beta_1\alpha_k\nabla f(x_k)^TB_k^{-1}\nabla f(x_k)\geq\frac{\beta_1C}{M}\|\nabla f(x_k)\|^2.
$$
A <telescoping sum> proves the displayed convergence of a nonnegative <series> and hence $\nabla f(x_k)\to0$. A uniform positive lower bound on the <eigenvalues> of $B_k$ is unnecessary for this conclusion. The conclusion is first-order stationarity; without further assumptions it does not assert convergence of the iterates or attainment of a <global minimum>.
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