a problem of upper limit

Question

Let $\{a_n\}$ be a sequence in $[-\infty,\infty]$. For $k=1,2,3,\ldots$, let $$b_k=\sup\{a_k,a_{k+1},\ldots\}$$ and $$\beta=\inf\{b_1,b_2,b_3,\ldots\}.$$ Then we say $\beta$ is the upper limit of $\{a_n\}$, named by $\beta=\limsup\limits_{n\to\infty} a_n$. From the above definition, we can easily get $$b_1\geq b_2\geq b_3\geq\cdots,$$ hence $b_k\to\beta$ when $k\to\infty$. And also, there is a subsequence $\{a_{n_i}\}$ of $\{a_n\}$ such that $a_{n_i}\to\beta$ when $i\to\infty$, and $\beta$ is the biggest number having this property. How can I construct such $\{a_{n_i}\}$ explicitly and prove the maximality of $\beta$?

Answer 1

Hint: when finite, $\inf S$ (resp. $\sup S$) is always a cluster point of the set $S$, meaning that every $\epsilon$-neighborhood of $\inf S$ (resp. $\sup S$) will contain an element of $S$. You can use this to construct your $\{a_{n_i}\}$ (provided $\beta$ is finite - if it's infinite, you should be able to see what to do).

Then, to show that $\beta$ is maximal, try a proof by contrapositive or contradiction.