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

We construct explicit examples of bounded sequences \ (\aₙ\₍=₁^\) in \ (R\) with prescribed behaviors for their accumulation properties. Specifically, we present one sequence whose set of subsequential limits S = \₊ a₍䂵 \{a₍䂵\ is a convergent subsequence of \aₙ\ \} has cardinality \ (₀\), the smallest infinite cardinality. We also construct a different example where the set of limit points T = \x R x is a limit point of \{aₙ\ \} 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.