2.7.E: Problems on Upper and Lower Limits of Sequences in \(E^{*}\) (Exercises)
Complete the missing details in the proofs of Theorems 2 and \(3,\) Corollary \(1,\) and Examples (a) and (b).
State and prove the analogues of Theorems 1 and 2 and Corollary 2 for
\(\underline{\lim} x_{n}\).
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?
\(\Rightarrow 4 .\) 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.]
\(\Rightarrow 5 .\) 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 \(0 \leq a<+\infty\).
Prove that
\[
\overline{\lim } x_{n}<+\infty\left(\underline{\lim} x_{n}>-\infty\right)
\]
iff \(\left\{x_{n}\right\}\) is bounded above (below) in \(E^{1}\).
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.]
\(\Rightarrow 8 .\) Prove that if \(p=\lim x_{n}\) in \(E^{1},\) then
\[
\underline{\lim} (x_{n} + y_{n}) = p + \underline{\lim} y_{n};
\]
similarly for \(\overline{L}\).
\(\Rightarrow 9 .\) 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
\[
\lim x_{n}=\inf _{n} x_{n}.
\]
\(\Rightarrow 10 .\) 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 \(\lim y_{n}=-\infty\).
Prove that if \(x_{n} \leq y_{n}\) for all \(n,\) then
\[
\underline{\lim} x_{n} \leq \underline{\lim} y_{n} \text{ and } \overline{\lim} x_{n} \leq \overline{\lim} y_{n}.
\]