= Solution
Let $\Delta_j=W_{t_j}-W_{t_{j-1}}$ for $1\le j\le n$. These increments are jointly normal, because each is obtained by evaluating the <isonormal Gaussian process> on an interval indicator. Indicators of distinct intervals are orthogonal in $L^2([0,\infty))$, so
$$
\operatorname{Cov}(\Delta_j,\Delta_k)=0\qquad(j\ne k).
$$
By <uncorrelated jointly Gaussian variables are independent>, \b[the increments on all these disjoint intervals are independent]. Reversing the sign of the first increment, as in the printed list, preserves this independence.
The three requested properties have now been obtained without an existence theorem for <Brownian motion>. One can also obtain continuous paths: the <Gaussian fourth moment> gives $\mathbb E|W_t-W_s|^4=3|t-s|^2$. The <Kolmogorov continuity theorem> therefore supplies a continuous modification on every finite time interval, which can be chosen consistently on the half-line. Modification preserves every finite-dimensional distribution and hence the independent Gaussian increments. This yields <Brownian motion> itself.
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