Edit: there's something wrong with the question.
I have got homework at Calculus and I can't prove this one:
Let $a_n>0$ be a bounded sequence and so that:
$\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1{\displaystyle\liminf_{n\to\infty}a_n}$
Prove that $a_n$ converges.
Generally, if we'd show that:
$\displaystyle\limsup_{n\to\infty}\frac 1 {a_n} = \frac 1 {\displaystyle\limsup_{n\to\infty} a_n}$
We're done because from plugging it in the given we'll see that
${\displaystyle\limsup_{n\to\infty} a_n} = {\displaystyle\liminf_{n\to\infty} a_n}$
And therefore $a_n$ converges. So firstly, from the given equation we can induce that:
$\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}, \displaystyle\liminf_{n\to\infty}a_n \neq 0$
Then we can easily prove that $\frac 1 {a_n}$ is bounded as well.
So, by B.W we can say that there exists a converged subsequence so that:
${\displaystyle\limsup_{n\to\infty}\ a_n}$ = ${\displaystyle\lim_{k\to\infty}\ a_{n_k}}$
From here we can say that the given applies also for subsequences of $a_n$ :
$\displaystyle\limsup_{k\to\infty}\frac 1 {a_{n_k}}=\frac 1 {\displaystyle\liminf_{k\to\infty}a_{n_k}}$
I didn't get anything out of that but that's what I've tried.
Thanks ahead!
