I'm taking real analysis and I still find this concept hard to understand:
If S is any bounded nonempty set of real numbers, $\sup(S)$ and $\inf(S)$ exist.
However, consider a bounded decreasing sequence $\{a_n\}$. Take $a_n= \frac 1 n$). Consider $\{b_n\}$ given by $b_n= \sup \{a_k: k \ge n \}$. We could prove $b_n$ was monotonic and bounded, thus converges. We wrote $\limsup a_n =\lim b_n$. In our case, $\lim b_n = \lim \frac 1 n = 0$.
However, $b_n$ is the superior of $\{a_n\}$, as the least upper bound of $a_n$, thus $b_n$ must be bigger or equal to $a_n$, which contradicts the result.
I searched online for limit superior but couldn't understand the notation well.
Could you help me to explain:
- What is the real definition of limit superior in $\mathbb{R}$?
- What went wrong, and the correct answer to the question?