Explicit examples of bounded sequences with countably infinite subsequential limits and limit points

We construct explicit examples of bounded sequences \(\{a_n\}ₙ₌₁^∞\) in \(R\) with prescribed behaviors for their accumulation properties. Specifically, we present one sequence whose set of subsequential limits S = →∞ anₖ ₖ\ is a convergent subsequence of ₙ\ \ has cardinality \(_0\), the smallest infinite cardinality. We also construct a different example where the set of limit points T = ∈ R x is a limit point of ₙ\ \ has cardinality \(_0\) as well. These examples illustrate that not only can \(S\) and \(T\) differ in structure, but that both sets can be countably infinite---a possibility not often emphasized in introductory analysis. This work contributes to a deeper understanding of the diversity of limiting behavior in sequences and highlights the subtle distinctions between subsequential limits and limit points.