Calculate supremum, infimum, lim sup and lim inf of $a_{n}=2-\frac{1}{n}$ for $n \in \mathbb{N} $
$\sup = 2-\left(\lim_{n\rightarrow\infty}\frac{1}{n}\right)=2-0 = 2$
$\inf = 2-\frac{1}{1}=2-1=1$
$\limsup_{n\rightarrow\infty}\left(2-\frac{1}{n}\right)=2-0=2$
$\liminf_{n\rightarrow\infty}$ I'm not sure about that but I would say it doesn't exist, it will be 2, same as $\limsup$.
Is it alright?
Best Answer
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, $\lim \inf_{n \rightarrow \infty} 2 - \frac{1}{n} = 2$.