= Solution
Write $m_l=\lfloor Nt_l\rfloor-\lfloor Nt_{l-1}\rfloor$ and $\phi(v)=\mathbb E e^{iv\xi}$. Since $\mathbb E\xi=0$ and $\mathbb E\xi^2=1$, the <characteristic function> expansion at zero is $\phi(v)=1-v^2/2+o(v^2)$. Indeed, $|e^{ix}-1-ix|\le x^2/2$ for real $x$, and $(e^{iv\xi}-1-iv\xi)/v^2\to-\xi^2/2$. The <dominated convergence theorem>, with dominating variable $\xi^2/2$, proves the expansion after taking expectations. Initially omit the endpoint interpolation terms. The integer-time increments involve disjoint sets of <independent> steps, so their joint <characteristic function> is
$$
\prod_{l=1}^k\phi(\lambda_l/\sqrt N)^{m_l}.
$$
Here $m_l/N\to t_l-t_{l-1}$. Taking the logarithm near $1$ gives
$$
\sum_lm_l\log\phi(\lambda_l/\sqrt N)
\longrightarrow-\frac12\sum_l\lambda_l^2(t_l-t_{l-1}).
$$
This proves the claimed limit for the integer-time approximation.
At a fixed time $t$, the omitted endpoint term is $R_N(t)=N^{-1/2}(Nt-\lfloor Nt\rfloor)\xi_{\lfloor Nt\rfloor+1}$, and $\mathbb E|R_N(t)|^2\le1/N$. A fixed linear combination of the errors in the $k$ increments therefore tends to zero in $L^1$, by the triangle inequality and <Cauchy-Schwarz inequality>, even when two endpoint terms share a step. Since $|e^{ix}-e^{iy}|\le|x-y|$, restoring interpolation changes the joint <characteristic function> by a quantity tending to zero. Hence
$$
\boxed{\varphi^N_{t_1,\ldots,t_k}(\lambda_1,\ldots,\lambda_k)
\longrightarrow \exp\left(-\frac12\sum_{l=1}^k\lambda_l^2(t_l-t_{l-1})\right).}
$$
The limit is the joint <characteristic function> of <independent> centered <normal> increments with the indicated variances. By <Lévy continuity theorem>, the increment vectors converge in distribution to <Brownian increment> vectors. The value vector is obtained by the fixed invertible triangular map of cumulative sums; thus the <continuous mapping theorem> gives convergence of $(S_{t_1}^N,\ldots,S_{t_k}^N)$ to $(B_{t_1},\ldots,B_{t_k})$. This proves every required <finite-dimensional distribution> limit. The finite <fourth moment> is not needed for this part; it will give tightness in part (b).
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