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If $\lim \sup _{n \rightarrow \infty} a_n=-\infty$, show that $\left(a_n\right)$ diverges to $-\infty$. If $\lim \sup _{n \rightarrow \infty} a_n=$ $+\infty$, show that $\left(a_n\right)$ has a subsequence that diverges to $+\infty$. What happens if $\liminf _{n \rightarrow \infty} a_n= \pm \infty$ ?

   If $\lim \sup _{n \rightarrow \infty} a_n=-\infty$, show that $\left(a_n\right)$ diverges to $-\infty$. If $\lim \sup _{n \rightarrow \infty} a_n=$ $+\infty$, show that $\left(a_n\right)$ has a subsequence that diverges to $+\infty$. What happens if $\liminf _{n \rightarrow \infty} a_n= \pm \infty$ ?

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Real analysis

N. L. Carothers 1st Edition

Chapter 1, Problem 25

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Step 1

The limit superior, \(\limsup_{n \to \infty} a_n\), is defined as the largest limit point of the sequence \((a_n)\). It can also be interpreted as the supremum of the set of subsequential limits. Show more…

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If $\lim \sup _{n \rightarrow \infty} a_n=-\infty$, show that $\left(a_n\right)$ diverges to $-\infty$. If $\lim \sup _{n \rightarrow \infty} a_n=$ $+\infty$, show that $\left(a_n\right)$ has a subsequence that diverges to $+\infty$. What happens if $\liminf _{n \rightarrow \infty} a_n= \pm \infty$ ?

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