From chapter 4 of Bulinskiy & Shiryaev's Theory of Random Processes (ISBN 5-9221-0335-0):
[Exercise 26] Let $W = \{W_t, t \geqslant 0 \}$ be a $m$-dimensional Brownian motion, where $m \geqslant 3$. Prove that $\|W_t\| \to \infty$ b.s. for $t \to \infty$. Prove that with probability $1$ the process $W$ will not return to point $0$ in any finite time. Explain why for $m=1$ these statements are not true. What does the following result look like in case $m=2$?
Solution:
Let $W = \{W_t, t \geqslant 0 \}$ be a $m$-dimensional Brownian motion in $\Bbb R^m$. Consider the process
$$\|W_t\| = \left( W_1^2 + \dots + W_m^2 \right)^{\frac12} = |W_t|$$
I tried to see the probability limit $ \lim\limits_{t \to \infty} \Bbb P \left( |W_t|^{-1} \right)$, it is equal to $0$. Therefore, $ \lim\limits_{t \to \infty}|W_t|^{-1} = 0 $ almost certainly. This means that almost certainly $|W_t| \to \infty$ for $t \to \infty$. Did I do the right thing?
Then I saw the idea where it was used that $|W_t|^{-1}$ is a super martingale, where $t \geqslant \delta > 0$, but I don't know how to prove it. And I also don't know what to do with special cases with $m=1$ and $m=2$.