Subsequences of real valued sequences and convergence

Convergence of real valued sequences is a classical subject of study for many mathematicians and it continues to be studied in recent years. Different types of convergence including ordinary, statistical, almost and ideal convergence, and related properties have been researched. In some studies, the relationship of a sequence and its subsequences regarding some type of convergence is investigated, using Lebesgue measure and Baire category. In this paper we revisit ordinary convergence and study the properties of subsequences of a real valued sequence using measure and category. We state and prove some simple theorems offering fresh insights into a classical subject.