Consider a bounded sequence $x_{n}$ of real numbers. I want to proof that the $limsup$ of a subsequence $x_{kn}$ with $k \in \mathbb{R}$ is smaller or equal to the $limsup$ of the mainsequence $x_{n}$ or in symbols:
$\limsup_{n\to\infty} x_{kn} \leq \limsup_{n\to\infty} x_{n}$.
I can see why this is correct, but don't know how to prove it. I think it is because $x_{kn}$ is a subsequence so if the mainsequence converges, so will the subsequence do, but I can't see how to fix it with the suprema.
Thanks in advance.