Let $\{x_n\}$ be a bounded sequence. Recall that a limit point of $\{x_n\}$ is the limit of some subsequence $\{x_{n_k}\}$ of $\{x_n\}$.

1. Show that if the sequence $\{x_n\}$ converges to $l\in\mathbb{R}$, every subsequence of $\{x_n\}$ also converges to $l$.
2. Show that the following are equivalent.
   • (a) $\{x_n\}$ is convergent.
   • (b) $\limsup x_n = \liminf x_n$.
   • (c) $\{x_n\}$ has a unique limit point.
3. Let $L$ be the set of limit points of ${x_n}$. Show that $\limsup x_n = \sup L$ and $\liminf x_n = \inf L$.