Let $(M,g)$ be a closed smooth Riemannian manifold and let $(N,h)$ be a compact smooth Riemannian manifold whose sectional curvature satisfies $\operatorname{sec}_h \leq 0$. For every smooth map $u_0: M \to N$, there exists a smooth harmonic map $u_\infty: M \to N$ that is homotopic to $u_0$.