Let $X$ be a [metric space](/page/Metric%20Space) with specified finite $2^{-k}$-nets $E_k$ for every $k$. Let $(x_n)$ be a sequence in $X$. Suppose branch-selection data are given: for each $k$ there is a point $e_k\in E_k$ and an infinite increasing sequence of indices $I_k(0)<I_k(1)<\cdots$ such that every $x_{I_k(j)}$ lies within $2^{-k}$ of $e_k$, and the index sequence for $k+1$ is a subsequence of the index sequence for $k$. Then there is a strictly increasing sequence $(n_m)$ such that for all $k$ and all $p,q\ge k+1$ one has $d(x_{n_p},x_{n_q})\le 2^{-k}$.