Real Analysis: Limit Superior and Bolzano Weierstrass

Prove the Bolzano-Weierstrass theorem: Every bounded sequence has a convergent subsequence.

Consider whether the converse of the Bolzano-Weierstrass theorem is true: If a sequence has a convergent subsequence, then it is bounded. Give an example of an unbounded sequence that has a convergent subsequence.

Prove that if a sequence is convergent, meaning it has a limit, then the lim sup of the sequence is equal to its limit.

Let the sequence $s_n$ be this sequence that repeats in a pattern of $0, 1, 2, 1, 0, 1, 2, 1$ and so on. What is the limit superior of this sequence?

Consider the sequence $t_n$ which approaches 3 with its odd terms and approaches 1 from above with its even terms. What is the limit superior of this sequence?

For the sequence $a_n$ whose nth term is $\frac{1 + (-1)^n n}{n}$, what are the limit superior and limit inferior of the sequence?

Let $b_n$ be the sequence $n \times \sin\left( \frac{\pi n}{2} \right)$. What are the limit superior and limit inferior of this sequence?

For a sequence $a_n$, determine the limit superior and limit inferior.