Let $B_t=(B_t^1,B_t^2)$ be a Brownian motion in $\mathbb{R}^2$. I wish to show that $P[B_t^2=0, t\in(0,\epsilon)]=1$ for all $\epsilon$; that is, a Brownian motion in $\mathbb{R}^2$ will always hit a line through the origin in some short time. For context, this is part of an attempted solution for Exercise 9.7c in Oksendal's SDE book, 6th edition.
My reason for believing this is that I know that one-dimensional Brownian motion equals zero infinitely many times in a short time interval, but I'm not sure how to adapt this.
Any help would be appreciated. Thanks!