= Solution
For times $t_1,\ldots,t_n<1$ and real coefficients $a_i$, the <linear combination> of coordinates is
$$
\sum_i a_iX_{t_i}=\int_0^{\max_i t_i}\left(\sum_i\frac{a_i(1-t_i)\mathbf1_{[0,t_i]}(s)}{1-s}\right)dB_s.
$$
The <Gaussianity of deterministic Brownian stochastic integrals> makes this a centered <normal random variable>. Indeed step-function approximations give <linear combinations> of independent normal increments, and the <Itô isometry> gives convergence in $L^2$; their <characteristic functions> converge to that of a normal variable with the limiting variance. Since every <linear combination> is normal, the coordinate vector is a <Gaussian random vector>. Appending the deterministic coordinate $X_1=0$ preserves that property, so $X$ is a centered <Gaussian process> on $[0,1]$.
For $0\le s\le t<1$, the <covariance> version of the <Itô isometry> gives
$$
\mathbb E[X_sX_t]=(1-s)(1-t)\int_0^s\frac{du}{(1-u)^2}
=(1-s)(1-t)\frac{s}{1-s}=s(1-t).
$$
At $t=1$ the <covariance> is zero because $X_1=0$, agreeing with the formula. Thus the <Brownian bridge covariance kernel> is
$$
\boxed{\Gamma(s,t)=\min(s,t)-st,\qquad\mathbb EX_t=0}.
$$
Its finite-dimensional distributions identify it as a <Brownian bridge>; continuity at the terminal endpoint is established next.
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