ANALISIS KRETERIA CAUCHY UNTUK KONVERGENSI BARISAN: STUDI TENTANG KEPERLUAN RUANG METRIK YANG LENGKAP

Abstract

The Cauchy criterion provides an intrinsic approach to determining the convergence of sequences without explicitly identifying their limits. However, this criterion does not always guarantee convergence when sequences are defined in metric spaces that lack completeness. This study aims to analyze the role of completeness of metric spaces in ensuring the validity of the Cauchy criterion and to examine its extension in generalized metric spaces. The research method employed is a qualitative study using a systematic literature review of classical textbooks and contemporary journal articles related to Cauchy sequences, complete metric spaces, and fixedpoint theory. The analysis shows that in complete metric spaces, every Cauchy sequence is convergent, making the Cauchy criterion a necessary and sufficient condition for convergence (Rudin, 1976; 57–58). In contrast, in incomplete metric spaces, there exist Cauchy sequences that fail to converge within the space. Furthermore, the study finds that in generalized metric spaces, such as b-metric spaces and fuzzy metric spaces, completeness remains a fundamental assumption, often characterized through fixed point theorems (Gregori et al., 2020; 2; Romaguera, 2023; 4). These findings confirm that completeness plays a central role in classical and modern mathematical analysis.