Let $\{a_n\}:\mathbb{N}\to\mathbb{R}$, and we define $E$ to be the set of all sub-sequential limits of $\{a_n\}$ as well as possibly the symbols $\infty$ and $-\infty$ if there are some sub-sequences of $\{a_n\}$ that go to $\infty$ / $-\infty$.
Can $E$ be empty? In particular, since $\limsup_{n \to \infty}a_n := \sup E$, does the upper limit of a sequence of real numbers is always defined? From what I found on the Wikipedia page on limit superior and limit inferior, the answer is yes and this have something to do with the fact that $\mathbb{R}$ is complete, but I couldn't find the complete proof, and I'm a bit surprised the issue isn't discussed on Rudin's principles of mathematical analysis.
Best Answer
Let $\sigma=\langle a_n:n\in\Bbb N\rangle$ be a sequence in $\Bbb R$. If $\sigma$ is unbounded, then it has a subsequence converging to $\infty$ or to $-\infty$, so $E\ne\varnothing$. If $\sigma$ is bounded, then by the Bolzano-Weierstrass theorem it has a convergent subsequence, so again $E\ne\varnothing$.