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Mathematics LibreTexts

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

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    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
    \[
    \underline{\lim} x_{n} \leq \underline{\lim} y_{n} \text{ and } \overline{\lim} x_{n} \leq \overline{\lim} y_{n}.
    \]