= Solution
The <Gaussian process> assumption and the covariance formula give a centered Gaussian increment with <variance>
$$
\mathbb E|\xi_t-\xi_s|^2=t+s-2\min(s,t)=|t-s|.
$$
For every finite $p>1$, a standard normal variable $Z$ therefore gives
$$
\|\xi_t-\xi_s\|_p=\|Z\|_p|t-s|^{1/2}.
$$
For any $0<\alpha<1/2$, choose $p>\max(2,(1/2-\alpha)^{-1})$. Then $\alpha<1/2-1/p$, so parts (a) and (b) give a <Hölder continuous function> of exponent $\alpha$ as an extension. To obtain one extension with every required exponent, take a countable sequence $\alpha_j\uparrow1/2$ and apply the construction with suitable $p_j$. Intersect the probability-one <events>. Their continuous extensions agree on the dense set $D$ and hence agree everywhere. For each smaller exponent choose $j$ with $\alpha<\alpha_j$; since $|t-s|\leq1$, the bound with $\alpha_j$ also implies the bound with $\alpha$. Thus
$$
\boxed{X\in C^{0,\alpha}([0,1])\quad\text{almost surely, simultaneously for every }0<\alpha<\tfrac12.}
$$
The exponent constants may depend on $\alpha$ and the sample path. This proves <Brownian Hölder regularity> for the continuous Gaussian extension without making an uncountable intersection of unrelated full-probability <events>.
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