Given following set $A_{n} = (-\infty, \frac{1}{n} + (-1)^n)$
I tried to do something like this:
Found a $\sup$ and $\inf$ of given set $$\sup A_{n} = \frac{1}{n} + (-1)^n$$ $$\inf A_{n} = -\infty$$
and then found limits: $$\lim_{n\rightarrow \infty}\sup A_{n} = \lim_{n\rightarrow \infty}\frac{1}{n} + (-1)^n $$ so I cannot compute previous limit
$$\lim_{n\rightarrow \infty}\inf A_{n} = -\infty$$
I feel I was wrong somewhere. Can somebody explain how should I do it or where I am wrong?
You have that $$ \sup A_n = \left\{ \begin{gathered} - 1 + \frac{1} {n}\,\,\,\,if\,\,\,n\,\,is\,\,\,odd \hfill \\ 1 + \frac{1} {n}\,\,\,if\,\,\,n\,\,is\,\,\,even \hfill \\ \end{gathered} \right. $$ therefore $$ \mathop {\lim }\limits_{n \to + \infty } \left( {\sup A_n } \right) = \left\{ \begin{gathered} - 1\,\,if\,\,\,n\,\,is\,\,\,odd \hfill \\ 1\,\,\,if\,\,\,n\,\,is\,\,\,even \hfill \\ \end{gathered} \right. $$