I am quite new to discrete and continuous stochastic processes. It seems there is something I don`t understand about definition of Brownian motion.
Let $\Omega, \mathcal{F}, \mathbb{P}$ be a probability space and $B_t$ be a standard continuous Brownian motion $B_t: \Omega \rightarrow C(\mathbb{R}^+, \mathbb{R})$.
Then the definition requires that $B_t \sim N(0,t)$ and $B_t - B_s \sim B_{t-s} \sim N(0,t-s)$. I don`t see how the first condition is compatible with the last. We know that $B_t \sim N(0,t)$ and $B_s \sim N(0,s)$ so $B_t - B_s \sim N(0,t+s)$ (seen as a sum of two normally distributed random variables) which seems to be in contradiction with independent increments. I guess I am missing something very basic but since I am new to this topic I cant see it. Thanks
update: based on answers below, yes it was indeed very basic, $B_t$ and $B_s$ are not independent as random variables so that is why the usual rule for sum of independent normally distributed random variables dont apply
Answer: The values of a Brownian motion path at different times are not independent. The variance of $B_t-B_s$ may at first seem like it should be $t+s$, but we can think of its variance being reduced due to $B_t$'s dependence on $B_s$.
Elaboration: We are only looking at a single Brownian motion process here, $B$. The value of $B$ at time $t$ is exactly its value at time $s$ plus some random fluctuation: $B_t(\omega)=B_s(\omega) + $ "the change in value from $s$ to $t$ for path realization $B(\omega)$". The part in quotes is declared to be independent from $B_s$ but also to be normally distributed with mean zero and variance equal to the length of time it covers: $t-s.$
Of course $B_t=B_s+(B_t-B_s)$ with $B_s$ and $B_t-B_s$ being independent. Given that $\text{Var}(B_t)=t$ and $\text{Var}(B_s)=s$, we can deduce that $\text{Var}(B_t-B_s)=t-s$.
If we were allowed to plug in distinct $\omega$'s to evaluate $B_t$ and $B_s$ independently, then the variance would be $t+s$.