Manuharawati, Muhammad Jakfar, Ahmad Taufik Hamzah
The convergence of sequences of real numbers in a metric space is one of the fundamental concepts studied in mathematical analysis. A sequence that approaches a real number x as its elements approach infinity is called a sequence that converges to x. In 1951, an extension of the concept of convergence, known as statistical convergence, was discovered. A sequence whose proportion of elements approaches a real number x as the elements approach infinity is said to be statistically convergent to x. The concept of convergence for sequences of real numbers has continued to develop. In 2000, a new concept of convergence called Ideal convergence was discovered. A sequence that is ideally convergent to a real number x means that the set of elements approaching x in the sequence is a member of an Ideal. This research employs an analytical method to explore the relationship between these three concepts of convergence - convergence, statistical convergence, and Ideal convergence - in the usual metric space ℝ. By analyzing specific examples and counterexamples, as well as proving related theorems, the study aims to provide a new approach for determining whether a sequence is convergent. One of the key theorems established is that if a sequence is convergent, then it is statistically convergent; however, the converse of this theorem does not hold. A statistically convergent sequence will be convergent if the sequence is monotonic. © 2025 Author(s).
Department of Mathematics, Universitas Negeri Surabaya, Surabaya, 60231, Indonesia
