
# 2.7.E: Problems on Upper and Lower Limits of Sequences in $$E^{*}$$ (Exercises)


Exercise $$\PageIndex{1}$$

Complete the missing details in the proofs of Theorems 2 and $$3,$$ Corollary $$1,$$ and Examples (a) and (b).

Exercise $$\PageIndex{2}$$

State and prove the analogues of Theorems 1 and 2 and Corollary 2 for
$$\underline{\lim} x_{n}$$.

Exercise $$\PageIndex{3}$$

Find $$\overline{\lim } x_{n}$$ and $$\underline{\lim} x_{n}$$ if
(a) $$x_{n}=c$$ (constant);
(b) $$x_{n}=-n$$ ;
(c) $$x_{n}=n ;$$ and
(d) $$x_{n}=(-1)^{n} n-n$$
Does $$\lim x_{n}$$ exist in each case?

Exercise $$\PageIndex{4}$$

$$\Rightarrow 4 .$$ A sequence $$\left\{x_{n}\right\}$$ is said to cluster at $$q \in E^{*},$$ and $$q$$ is called its cluster point, iff each $$G_{q}$$ contains $$x_{n}$$ for infinitely many values of $$n$$.
Show that both $$\underline{L}$$ and $$\overline{L}$$ are cluster points $$(\underline{L} \text { the least and } \overline{L} \text { the }$$ largest).
[Hint: Use Theorem 2 and its analogue for $$\underline{L}$$ .
To show that no $$p<\underline{L}$$ (or $$q>\overline{L} )$$ is a cluster point, assume the opposite and find a contradiction to Corollary 2.]

Exercise $$\PageIndex{5}$$

$$\Rightarrow 5 .$$ Prove that
(i) $$\overline{\lim} \left(-x_{n}\right)=-\underline{\lim} x_{n}$$ and
(ii) $$\overline{\lim} \left(a x_{n}\right)=a \cdot \overline{\lim } x_{n}$$ if $$0 \leq a<+\infty$$.

Exercise $$\PageIndex{6}$$

Prove that
$\overline{\lim } x_{n}<+\infty\left(\underline{\lim} x_{n}>-\infty\right)$
iff $$\left\{x_{n}\right\}$$ is bounded above (below) in $$E^{1}$$.

Exercise $$\PageIndex{7}$$

Prove that if $$\left\{x_{n}\right\}$$ and $$\left\{y_{n}\right\}$$ are bounded in $$E^{1},$$ then
$\overline{\lim } x_{n}+\overline{\lim } y_{n} \geq \overline{\lim }\left(x_{n}+y_{n}\right) \geq \overline{\lim } x_{n}+\underline{\lim} y_{n} \geq \underline{\lim} (x_{n} + y_{n}) \geq \underline{\lim} x_{n} + \underline{\lim} y_{n}.$
[Hint: Prove the first inequality and then use that and Problem 5$$(\mathrm{i})$$ for the others.]

Exercise $$\PageIndex{8}$$

$$\Rightarrow 8 .$$ Prove that if $$p=\lim x_{n}$$ in $$E^{1},$$ then
$\underline{\lim} (x_{n} + y_{n}) = p + \underline{\lim} y_{n};$
similarly for $$\overline{L}$$.

Exercise $$\PageIndex{9}$$

$$\Rightarrow 9 .$$ Prove that if $$\left\{x_{n}\right\}$$ is monotone, then $$\lim x_{n}$$ exists $$i n E^{*} .$$ Specifically, if $$\left\{x_{n}\right\} \uparrow,$$ then
$\lim x_{n}=\sup _{n} x_{n},$
and if $$\left\{x_{n}\right\} \downarrow,$$ then
$\lim x_{n}=\inf _{n} x_{n}.$

Exercise $$\PageIndex{10}$$

$$\Rightarrow 10 .$$ Prove that
(i) if lim $$x_{n}=+\infty$$ and $$(\forall n) x_{n} \leq y_{n},$$ then also $$\lim y_{n}=+\infty,$$ and
(ii) if $$\lim x_{n}=-\infty$$ and $$(\forall n) y_{n} \leq x_{n},$$ then also $$\lim y_{n}=-\infty$$.

Exercise $$\PageIndex{11}$$

Prove that if $$x_{n} \leq y_{n}$$ for all $$n,$$ then
$\underline{\lim} x_{n} \leq \underline{\lim} y_{n} \text{ and } \overline{\lim} x_{n} \leq \overline{\lim} y_{n}.$