Let $(X,d)$ be a complete metric space, and let $E:X\to(-\infty,\infty]$ be lower semicontinuous. Assume that, for every $c\in\mathbb R$, the sublevel set
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\begin{align*}
\{x\in X:E(x)\le c\}
\end{align*}
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is compact in $X$. Fix $x_0\in X$ such that $E(x_0)<\infty$.
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For each $\tau>0$, define a sequence $(x_k^\tau)_{k\ge 0}$ in $X$ by $x_0^\tau=x_0$ and, for every $k\ge 0$, by choosing $x_{k+1}^\tau\in X$ such that
whenever $k\in\mathbb N\cup\{0\}$ and $t\in[k\tau,(k+1)\tau)$.
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If $(\tau_j)_{j\ge 1}$ is any sequence in $(0,\infty)$ such that $\tau_j\to 0$, then there exist a subsequence $(\tau_{j_\ell})_{\ell\ge 1}$ and a continuous curve $x:[0,\infty)\to X$ with $x(0)=x_0$ such that, for every $T>0$,
as $\ell\to\infty$. Any such locally uniform limit curve is called a minimizing movement for $E$ starting from $x_0$ along the vanishing time steps $(\tau_{j_\ell})_{\ell\ge 1}$.