for bounded series $x_n$, prove that $\limsup(1/x_n) = 1/ \liminf( x_n)$

Question

Stuck on a proof about series: i know that $ 0 < a \leq x_n \leq b, $

$a$ and $b$ are real numbers (not infinite)

and from this it follows that $1/x_n$ is also bounded by $1/b$ and $1/a,$ which are positive.

How can i prove that $ \limsup( 1/x_n ) = 1 / \liminf(x_n) ?$

I know that $\limsup(x_n) = -\liminf(- x_n) $

and by Bolzano-Weierstraas every bounded set has a converging subset.. But i can't seem to think of a solution to this one.

Thanks in advance!