Let $(X,d)$ be a complete metric space, let $\sigma$ be a Hausdorff topology on $X$, let $T>0$, and let $\mathcal F:X\to(-\infty,\infty]$ be a proper lower semicontinuous functional. Let $|\partial\mathcal F|:X\to[0,\infty]$ denote the local metric slope of $\mathcal F$. Define the $\sigma$-relaxed slope $G:X\to[0,\infty]$ by
with the convention that the infimum over an empty set is $+\infty$. Assume that $G$ is a strong upper gradient for $\mathcal F$.
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Let $\rho_0\in X$ satisfy $\mathcal F[\rho_0]<\infty$, and let $(\tau_n)_{n\in\mathbb N}$ be a sequence in $(0,\infty)$ such that $\tau_n\downarrow0$. For each $n\in\mathbb N$, suppose that there exists a sequence $(\rho_{\tau_n,k})_{k\ge0}$ in $X$ such that $\rho_{\tau_n,0}=\rho_0$ and, for every $k\ge1$,