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

We construct explicit examples of bounded sequences \(\{a_n\}_{n=1}^\infty\) in \(\mathbb{R}\) with prescribed behaviors for their accumulation properties. Specifically, we present one sequence whose set of subsequential limits $S = \{\displaystyle\lim_{k\to\infty} a_{n_k} \mid \{a_{n_k}\}$ is a convergent subsequence of $\{a_n\} \}$ has cardinality \(\aleph_0\), the smallest infinite cardinality. We also construct a different example where the set of limit points $T = \{x \in \mathbb{R} \mid x$ is a limit point of $\{a_n\} \}$ has cardinality \(\aleph_0\) 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.