I was looking at some proprieties of numerical sequence on extended real line but I'm stuck on a definition I found on Rudin textbook.
Here I have sequence $\,\{a_n\}$ defined on $[-\infty ,+\infty]$ and I want to figure out what's$$\beta=\lim_{n \to \infty}sup \,a_n$$ I have no problem with sequences defined on (-$\infty ,+\infty$) but if I take a sequence that diverges for some n , like $1\over n-2$ I found (according to definition) that $\beta=0$ (in this case)
Am I wrong? There's something that I miss?
The limit superior and limit inferior have very simple definitions, once you just write them out.
Consider $\{a_n\}$, a sequence. Take the set of all limit points of $a_n$ i.e. the limits of convergent subsequences of $a_n$. The infimum of this set, is $\liminf a_n$, and the supremum of that set is $\limsup a_n$.
So, all you have to do is focus on this set. In the first example you have given, $a_n = \frac 1{n-2}$, and we see that this sequence converges, and therefore has only one limit point, namely zero. So, $\limsup a_n = \liminf a_n = 0$.
Suppose it turns out that your set of limit points is empty (for example, in the sequence $1,2,3,4,...$), then automatically your limit superior would be $+\infty$ and your limit inferior would be $- \infty$. Unfortunately, the example $n \sin n$ does not seem easy to decide, since it seems difficult to predict it's limit points (although it surely does not converge). The sequence $\sin n$ itself is somewhat notorious : it's set of limit points is all of $[-1,1]$, and hence it's limit superior and inferior would be $1$ and $-1$ respectively.