On a variant of Schu's lemma

TL;DR

This paper identifies a long-standing error in the literature where 'lim' was incorrectly replaced by 'limsup' in a key lemma, affecting multiple papers over two decades.

Contribution

It clarifies the mistake in Schu's lemma and highlights the propagation of this incorrect statement in the mathematical community.

Findings

The incorrect substitution has been propagated in over 20 years of research.

The earliest paper with the mistake dates back to 2002.

The paper emphasizes the importance of precise mathematical statements.

Abstract

In this note, we demonstrate that an incorrect statement has been propagated in multiple papers, stemming from the substitution of ``lim'' with ``limsup'' for a sequence in Lemma 1.3 of the paper [J. Schu: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull.\ Austral.\ Math.\ Soc.\ 43 (1991), 153--159]. This occurred over a span of more than 20 years, with the earliest paper we identified using this incorrect statement dating back to 2002.