Question

Prove the characterization of lim sup given above. That is, given a bounded sequence $\left(a_n\right)$, show that the number $M=\lim \sup _{n \rightarrow \infty} a_n$ satisfies ( $$ ) and, conversely, that any number $M$ satisfying () must equal $\lim \sup _{n \rightarrow \infty} a_n$. State and prove the corresponding result for $m=\liminf _{n \rightarrow \infty} a_n$.

   Prove the characterization of lim sup given above. That is, given a bounded sequence $\left(a_n\right)$, show that the number $M=\lim \sup _{n \rightarrow \infty} a_n$ satisfies ( $*$ ) and, conversely, that any number $M$ satisfying (*) must equal $\lim \sup _{n \rightarrow \infty} a_n$. State and prove the corresponding result for $m=\liminf _{n \rightarrow \infty} a_n$.

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Real analysis

N. L. Carothers 1st Edition

Chapter 1, Problem 26

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Step 1

The limit superior is defined as: \[ M = \limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{k \geq n} a_k. \] This means that \(M\) is the largest value that the sequence approaches infinitely often. Show more…

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Prove the characterization of lim sup given above. That is, given a bounded sequence $\left(a_n\right)$, show that the number $M=\lim \sup _{n \rightarrow \infty} a_n$ satisfies ( $*$ ) and, conversely, that any number $M$ satisfying (*) must equal $\lim \sup _{n \rightarrow \infty} a_n$. State and prove the corresponding result for $m=\liminf _{n \rightarrow \infty} a_n$.

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