Let $X_k$ be independent and identically distributed random variables with mean $0$ and variance $1.$ Define for $0\leq t \leq1, Z_n(t) = \sum_{k=1}^{\left \lfloor{nt}\right \rfloor }X_k$. Show that for each finite partition $0\leq t_1 \lt \dotsb \lt t_N\leq 1$ the vector $Z_n = (Z_n(t_1),...,Z_n(t_N))$ converges in distribution to a $N$-dimensional Gaussian random vector with mean $0$ and covariance matrix $C$, with $C_{i,j}=\min(t_i, t_j)$.
For the covariance we have (after a quick calculation) $$Cov(Z_n(t_i), Z_n(t_j))=\dfrac{1}{n}\left \lfloor{n\cdot\min(t_i, t_j)}\right \rfloor\to\min(t_i, t_j)\text{ as }n\to\infty.$$
Now the part where I am stuck is the convergence. If the $Z_n$ were independent of $t$ this would be a quick application of the multivariate central limit theorem. I am struggling with finding an adaption to still be able to apply the CLT.
You need to use the fact that $X_n = (X_n^1, X_n^2, \dots, X_n^m) \to X =(X^1, X^2, \dots, X^m)$ in distribution if and only if $<\alpha, X_n> \to <\alpha, X>$ in distribution for each $\alpha \in \mathbb{R}^m$ ($<\cdot,\cdot>$ is scalar multiplication) and apply ordinary CLT.