I'm referencing from Leimkuhler and Matthews' Molecular Dynamics textbook.
From my understanding, a Wiener process $W(t)$ is essentially a random walk process at the diffusion limit $\delta x^2\approx\delta t$ and time limit $\delta t\to0$. We can solve for a Wiener process from $0$ to $t$ with the algorithm $W_{n+1} = W_n + \delta x$, where $\delta x$ is a vector satisfying $\vert\!\vert\delta x\vert\!\vert = \sqrt{\delta t}$ and so $\vert\!\vert dW\vert\!\vert$ is fixed as well. Another way to phrase this is $$\delta x = \hat{R} = \dfrac{R}{\vert\!\vert R\vert\!\vert},$$ where $R$ is a vector sampled iid from a normal distribution.
However, most numerical methods beyond chapter 6 imply that $dW = \sqrt{\delta t}R$. This implies that $\vert\!\vert dW\vert\!\vert$ can take on any nonnegative value.
So how did we transition from $$W_{n+1} = W_n + \sqrt{\delta t}\hat{R}$$ to $$W_{n+1} = W_n + \sqrt{\delta t}R?$$
I must have missed something.