Define the sequence $(a_n)$ as following $a_1=1$ for $n\geq 1$ $a_{n+1}=(-1)^n(1/2)(|a_n|+2/|a_n|)$ which of the following
(1) $\lim \sup a_n=\sqrt{2}$
(2) $\lim \inf a_n=-\infty$
(3) $\lim a_n=\sqrt{2}$
(4) $\sup a_n =\sqrt{2}$
I am not getting how to obtain $\lim \sup$ and $\lim \inf$... I tried getting some terms if the sequence $a_1=1,a_2=-\frac{3}{2},a_3=\frac{17}{12}..$ but unable to conclude anything ...just all the terms are less then 2... only this I can conclude
Thanks in advance!!