Question

If $\left(a_n\right)$ is convergent, show that $\liminf \operatorname{inc}_n a_n=\limsup \sin _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty} a_n$.

   If $\left(a_n\right)$ is convergent, show that $\liminf \operatorname{inc}_n a_n=\limsup \sin _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty} a_n$.

Real analysis

N. L. Carothers 1st Edition

Chapter 1, Problem 23

Instant Answer

Step 1

- $\limsup_{n \to \infty} a_n$ is the limit superior of the sequence $(a_n)$, defined as $\lim_{n \to \infty} (\sup_{m \geq n} a_m)$. - $\lim_{n \to \infty} a_n$ is the limit of the sequence $(a_n)$, which exists if the sequence is convergent. Show more…

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If $\left(a_n\right)$ is convergent, show that $\liminf \operatorname{inc}_n a_n=\limsup \sin _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty} a_n$.

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