Given a sequence $\left(a_n\right)$ of real numbers, let $\mathcal{S}$ be the set of all limits of convergent subsequences of $\left(a_n\right)$ (including, possibly, $\pm \infty$ ). For example, it follows from Exercise 27 that $\lim \sup _{n \rightarrow \infty} a_n$ and $\lim _{n \rightarrow \infty} a_n$ are both elements of $\mathcal{S}$. Show that, in fact, $\lim \sup _{n \rightarrow \infty} a_n=\sup \mathcal{S}$ and $\liminf _{n \rightarrow \infty} a_n=\inf \mathcal{S}$.