If $B\in \mathcal{B}([0,\infty))$ and we assume it is bounded i.e. there exists a $K > 0$ s.t. $B \subseteq [0,K]$.
Then is it true that $B \in \mathcal{B}([0,K])$?
I have some trouble proving this but I think it is true and shouldn't be too difficult to show.
I have tried using the definition of the borel sigma algebra as the smallest sigma algebra containing the open sets but this didn't lead anywhere. Maybe one could show it using an equivalent generating system?