Limit superior and limit inferior of a sequence

Question

I have trouble understanding the concepts limit superior and inferior using the traditional definition. I have to solve a large problem and I'm not sure of how to do it. I will thank you for any hints or references of books or something where I can find any help:

Exercise 2.6.4. Given a sequence $(x_n)_{n=1}^\infty$ defined in $\mathbb{R}$, number $x^* \in \mathbb{R}$ is said to be its limit superior if $$\forall \varepsilon > 0, \exists n^* \in \mathbb{N} : \forall n > n^* , x_n < x^* + \varepsilon$$ and $$\forall \varepsilon > 0, \forall n \in \mathbb{N}, \exists n_0 > n : x_{n_0} > x^* − \varepsilon.$$ Number $x_* \in \mathbb{R}$ is the sequence’s limit inferior if $$\forall \varepsilon > 0, ∃n_* \in \mathbb{N} : \forall n > n_*, x_n > x_* − \varepsilon$$ and $$\forall \varepsilon > 0, \forall n \in \mathbb{N}, \exists n_0 > n : x_{n_0} < x_* + \varepsilon.$$ When they exist, these numbers are denoted, respectively, as $\operatorname{limsup}_{n\to\infty} x_n = x^*$ and $\operatorname{liminf}_{n\to\infty} x_n = x_*$.

1. Does the existence of the limit superior of a sequence guarantee that its limit inferior also exists?
2. Argue that if $\operatorname{limsup}_{n\to \infty} x_n = x^*$, then there exists a subsequence $(x_{n_m})$ of $(x_n)$ that converges to $x^*$.
3. Argue that, when they both exist, $\operatorname{limsup}_{n\to\infty} x_n \ge \operatorname{liminf}_{n\to\infty} x_n$.
4. Give an example of a sequence for which the previous inequality is strong.
5. Argue that if $\lim_{n\to\infty} x_n = x$, then $\operatorname{limsup}_{n\to\infty} x_n = x$.
6. Argue that if $$\operatorname{limsup}_{n\to\infty} x_n = x = \operatorname{liminf}_{n\to\infty} x_n,$$ then $\lim_{n\to \infty} x_n = x$.

Answer 1

There are already questions out there which give some equivalent definitions and explanations about $\limsup$ and $\liminf$, for example here.

Personally, I like to think that the sequence eventually becomes "wedged" between the limsup and liminf, as illustrated in the picture below. The picture should give you some good intuition for your questions 2-6.