Calculate supremum, infimum, lim sup and lim inf of $a_{n}=2\frac{1}{n}$ for $n \in \mathbb{N} $
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VictorZurkowski already answered correctly in his comment.
For the concrete example: \begin{equation} \inf_{n \geq k} \, 2  \frac{1}{n} = 2  \frac{1}{k} \end{equation} So taking $\lim_{k \rightarrow \infty} 2  \frac{1}{k} = 2$.
Thus, $\lim \inf_{n \rightarrow \infty} 2  \frac{1}{n} = 2$.
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cnmesr
Updated on August 01, 2020Comments

cnmesr over 3 years
Calculate supremum, infimum, lim sup and lim inf of $a_{n}=2\frac{1}{n}$ for $n \in \mathbb{N} $
$\sup = 2\left(\lim_{n\rightarrow\infty}\frac{1}{n}\right)=20 = 2$
$\inf = 2\frac{1}{1}=21=1$
$\limsup_{n\rightarrow\infty}\left(2\frac{1}{n}\right)=20=2$
$\liminf_{n\rightarrow\infty}$ I'm not sure about that but I would say it doesn't exist, it will be 2, same as $\limsup$.
Is it alright?

VictorZurkowski over 7 yearsYou are correct. This is general, in the sense that the limit exists iff lim sup and lim inf are equal (lim sup and lim inf allways exist, but they may not be finite).


cnmesr over 7 yearsDoes that mean lim inf = 2 too? Or I better say doesn't exist?

ChrisT over 7 yearslim inf = 2, as VictorZurkowski already mentioned.