Let $(X,d)$ be a compact [metric space](/page/Metric%20Space), and let $f: X \to X$ be a continuous map. Define $f^0 := \operatorname{id}_X$ and $f^{n+1} := f \circ f^n$ for every integer $n \ge 0$. Then for every $x \in X$, there exist a point $y \in X$ and a strictly increasing sequence $(n_k)_{k \in \mathbb N}$ of nonnegative integers such that

\begin{align*} \lim_{k \to \infty} d(f^{n_k}(x), y) = 0. \end{align*}

Equivalently, the forward orbit sequence $(f^n(x))_{n \ge 0}$ has an accumulation point in $X$.