 Solution

For times $t_1,\dots ,t_n<1$ and real coefficients $a_i$, the linear combination of coordinates isThe 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 givesAt $t=1$ the covariance is zero because $X_1=0$, agreeing with the formula. Thus the Brownian bridge covariance kernel isIts finite-dimensional distributions identify it as a Brownian bridge; continuity at the terminal endpoint is established next.