Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 29K / a / Solution

A Brownian motion starts at zero, has continuous paths, and has independent increments with $W_t-W_s\sim N(0,t-s)$. The stated covariance makes every finite collection jointly Gaussian; disjoint increments have zero covariance and hence are independent. Their variances are interval lengths, proving all defining properties.