= Solution
Standard one-dimensional <Brownian motion> is a real process with $B_0=0$, continuous paths almost surely, and increments $B_t-B_s\sim N(0,t-s)$ <independent> of the past at time $s$, for $0\le s<t$. These conditions determine its entire <finite-dimensional distribution> family.
The <Wiener theorem> asserts that such a process exists, equivalently that there is a unique <Wiener measure> on $C_0([0,\infty),\mathbb R)$ under which the coordinate process has these properties. To construct it, prescribe centered multivariate <normal> laws with <covariance> $\mathbb E B_sB_t=\min(s,t)$. This <covariance> is <positive semidefinite> because
$$
\sum_{i,j}c_ic_j\min(t_i,t_j)=\int_0^\infty\left(\sum_i c_i\mathbf1_{[0,t_i]}(u)\right)^2du\ge0.
$$
The laws are consistent under removing coordinates, so the <Kolmogorov extension theorem> constructs the process. <Normal> increments have <fourth moment> $\mathbb E|B_t-B_s|^4=3|t-s|^2$. The <Kolmogorov continuity theorem> therefore supplies a continuous modification, retaining all the prescribed finite-dimensional laws and hence the <independent> <normal> increments. Apply this on every compact time interval, taking consistent versions; continuity and equality at rational times make them agree on overlaps. The resulting law on continuous-path space is unique because evaluations at rational times generate its <Borel sigma-algebra>. This is a proof sketch of the existence theorem, <independent> of the random-walk limit in Question 5.
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