In my lecture notes there is an (somewhat guided) exercise where one is supposed to show that the paths of a two dimensional Brownian motion are dense in $\mathbb{R}^2$. I came to a point where I am not sure how to proceed:
Let $B = (B_t)_{t \geq 0}$ denote a $\mathbb{R}^2$-valued Brownian motion starting at $x \in \mathbb{R}^2\setminus \{0\}$. We consider the stopping times $\tau_r = \inf\{t \geq 0: |B_t| = r\}$ for $r \in \mathbb{R}_+$. I proceeded to show that $\tau_0$ is a predictable stopping time and that $M := \log|B|$ is a local martingale on $[0,\tau_0)$ (i.e. for an announcing sequence $\sigma_n$ of $\tau_0$, the process $M^{\sigma_n}$ is a local martingale). I showed this using Ito's formula. We have $$ M_t = \log|B_t| = \log|x| + \int_0^t B_s' \frac{dB_s}{|B_s|^2} $$ and since the $ds$ term vanishes, we indeed have a local martingale (on $[0,\tau_0)$). For $r < R$ satisfying $0<r\leq|x|\leq R$,we have (using optional stopping) $$ \log|x| = \mathbb{E}[M^{\tau_r \land \tau_R}] = \log r \cdot\mathbb{P}[\tau_r < \tau_R] + \log R \cdot(1 - \mathbb{P}[\tau_r < \tau_R]). $$ This yields $\mathbb{P}[\tau_r < \tau_R] = \frac{\log R - \log|x|}{\log R - \log r}$.
Now, I am supposed to proceed to show that $\mathbb{P}[\lim \inf_{t \to \infty} |B_t| = 0] = 1$. Intuitively this is clear to me: Letting $R \to \infty$ in the above expression, we have $\mathbb{P}[\tau_r < \infty] = 1$. Therefore $B$ can come arbitrarily close to 0 in finite time. 'Some time later' $B$ is at some other point $y \in \mathbb{R}^2$. Since the Brownian motion is 'memoryless', one can argue that $B$ must come arbitrarily close to 0 again and so on. This ultimately should yield the desired result. However I am not sure how I can argue rigorously.
Additionally, from $\mathbb{P}[\lim \inf_{t \to \infty} |B_t| = 0] = 1$ I should conclude that that the paths of $B$ are dense in $\mathbb{R}^2$, which I am not sure about either. Maybe I get the idea for this, when I have shown that $\mathbb{P}[\lim \inf_{t \to \infty} |B_t| = 0] = 1$.
So my question is how to argue rigorously that $\mathbb{P}[\lim \inf_{t \to \infty} |B_t| = 0] = 1$?
Thanks in advance!