Why do we use linear interpolation in the contruction of Brownian motion?

Question

I am familiar with Lévy's construction of Brownian motion as the limit of linear interpolations of gaussian values. But could we define a stochastic process in one shot as $$ B_t := \sum_{\mathcal{D} \ni d<t} 2^{-(|d|+1)}Z_d, $$ Where $\mathcal{D} $ are the dyadic numbers and index a family of iid. standard normal gaussians, and $|d|$ is the length of the dyadic expansion of $d$, and then show that $B$ is a Brownian motion ? I feel like this formula could work, because the variances at $t$ sum to $t$, but I'm not sure how to proceed for a proof. I think I'd need some kind of result that the series of Gaussian RVs is again Gaussian, which I'm not sure about.