Random walk converging to Brownian Motion for $t \in [0, \infty)$

Question

If we define $X_i$ as a Bernoulli random variable with P(X=1) = P(X=-1) = 0.5. E[X_i] = 0, Var(X_i) = 1. As i have understood, by applying the Donsker's theorem, $t \in [0, 1]$, $B_N(t) \frac{1}{\sqrt{N}} \sum_{i=0}^{Nt} X_i$ converges to the Wiener process in distribution as $N \rightarrow \infty$. How can I extend t over $[0, \infty)$? Any hints are apprecitated