Toeplitz lemma in geodesic metric space and convergence of inductive means

In this article, we study the Toeplitz lemma for inductive means in a geodesic metric space, and by using the Toeplitz lemma, we prove the Cesàro theorem for inductive means. Also, we study an asymptotic property for Hansen's inductive geometric mean, and as an application of the Toeplitz lemma, we prove a convergence of Hansen's inductive mean.