I know that the increment $B_{t+u}-B_t$ is independent of $B_s$ for $s<t$, so it is clear that $B_{t+u}-B_t$ is independent of $B_{s+l}-B_s$ where $l<t-s$, i.e. two non-overlapping increments are independent. But is the same true for $B_{t+u}-B_t$ and $B_{t}-B_s$, i.e. when they share a single point? My gut feeling says yes, as, in some way, this single point is insignificant, but I cannot seem to find a sound argument.
Thank you very much in advance for the help!
Surely. One way of proving this is to write $$Ee^{ia(B_{t+u}-B_{t+\epsilon})+ib(B_t-B_s)}= Ee^{ia(B_{t+u}-B_{t+\epsilon})}Ee^{ib(B_t-B_s)}.$$ and let $\epsilon \to 0$. [Use DCT on left side].
More generally, independence is preserved under almost sure limits.