Prove that $\limsup a_n = \sup P$ and $\liminf a_n = \inf P$, where $P$ is the set of limit points

Let $\{a_n\}$ be a bounded sequence of real numbers and let $P$ be the set of limit points of $\{a_n\}$. Prove that $\limsup a_n = \sup P$ and $\liminf a_n = \inf P$

my work:

Since ${a_n}$ is bounded, then there is an $M$ such that ${a_n} \leq M$ for all $n$. By Bolzano-Weierstrass theorem, ${a_n}$ must have a subsequence that converges (in this case to $P$).
Then, $P \leq M$.
$\liminf {a_n} \leq \limsup {a_n} \leq M$.
Since $P \leq M$, then $\liminf {a_n} \leq \inf P \leq \sup P \leq\limsup {a_n} \leq M$.

I'm not sure how to finish though, or if what I've put is correct

Help is greatly appreciated!

• $P$ is a set, one which may contain more than one point. What do you mean when you say that $a_n$ converges to $P$? – Omnomnomnom Nov 18 '13 at 3:05
• subsequences which converges to a set of limit points P. – starter1011 Nov 18 '13 at 3:08
• Sequences don't "converge" to a set of points. If a sequence converges, it converges to a unique limit. – Omnomnomnom Nov 18 '13 at 3:13
• that's why im saying that a subsequence of a_n converges to a certain element of the set of limit points P. – starter1011 Nov 18 '13 at 3:15
• I see, much clearer now, thank you. That's not how things are usually said, though. Honestly, I'm having a hard time of understanding your line of thought. – Omnomnomnom Nov 18 '13 at 3:22

1. If $a_{n_k}\to a \in P$, then $\liminf a_n \leq a \leq \limsup a_n$.
2. There is a subsequence that converges to $\liminf a_n$, and one that converges to $\limsup a_n$.
From there, we may conclude that since the liminf and limsup are lower and upper bounds of $P$ that are elements of $P$, they must also be the greatest lower bound and least upper bound.
• I'm sorry if I've offended you. Do you understand why if the lim inf is both a lower bound of $P$ and a member of $P$, it must also be the inf of $P$? – Omnomnomnom Nov 18 '13 at 3:39
• I do see what you mean. I'm wondering though, what is your definition of $\limsup$? Depending on the definition you start with, this question is either a trivial application of a definition (as you suggest) or something a bit more subtle. Is your definition $\limsup a_n = \lim_{n\to \infty} \sup_{k>n}\{a_n\}$? – Omnomnomnom Nov 18 '13 at 4:02