= Solution
Let $(B_t)_{0\leq t\leq1}$ be standard <Brownian motion> with its continuous version, and define
$$
\boxed{Z_t=B_t-tB_1.}
$$
Every finite vector is a linear transform of a Gaussian Brownian vector, so $Z$ is a centred <Gaussian process>. Using $\mathbb E[B_sB_t]=\min(s,t)$,
$$
\mathbb E[Z_sZ_t]
=\min(s,t)-t\,s-s\,t+st
=\min(s,t)-st.
$$
For $s\leq t$, this is $s(1-t)$, the prescribed <Brownian bridge covariance kernel>. Its paths are continuous on the probability-one event of Brownian continuity, because $t\mapsto tB_1$ is continuous on every sample. The construction also gives $Z_0=Z_1=0$. Thus it realizes a <Brownian bridge> with <almost surely> continuous paths, and part (i) identifies its process law.
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