Let $f:[a,b]\to\mathbb{R}$ be a uniformly [continuous function](/page/Continuous%20Function) on a located compact interval, with a displayed modulus of [uniform continuity](/page/Uniform%20Continuity). Suppose there are nested rational intervals $[u_k,v_k]\subseteq[a,b]$ such that $u_k\le u_{k+1}\le v_{k+1}\le v_k$, $v_k-u_k\le 2^{-k}$, $f(u_k)\le 2^{-k}$, and $f(v_k)\ge -2^{-k}$ for every $k$. Then the midpoints of these intervals determine a Cauchy real $z\in[a,b]$ and $f(z)=0$.