Question: Suppose that ($a_n$) is a sequence such that L = lim sup ($a_n$) is a real number. Then we know that, for any number $M$ $>$ $L$, there are only finitely many integers $n$ for which ($a_n$) $>$ $M$. Show by means of an example that it is possible to have ($a_n$) $>$ $L$ for infinitely many n.
I don't really understand what the question means, any further explanation would be really appreciated. There is a Proposition in my textbook that I think may be helpful for this question, but I am not sure.
- Proposition 2.20
(a) If $L$ = $lim$ $sup$ ($a_n$) then
- for each $ε$ $>$ 0 the inequality ($a_n$) $>$ $L$ + $ε$ holds for only finitely many n and
- For each $ε$ $>$ 0 the inequality ($a_n$) $>$ $L$ - $ε$ holds for infinitely many n
Any help would be really appreciated
Thanks a lot