This is a proof verification. I would be grateful if you could point out any problems.
Let $W=\{W_t\}_{t \ge 0}$ be a standard two dimensional Brownian motion starting at the origin. For $r>0$ and $R>0$, we define $\sigma_{r}=\inf \{t>0 \mid B_t \in \{r\} \times \mathbb{R}\}$ and $\tau_R=\inf \{t>0 \mid B_t \notin B(R)\}$, where $B(R)$ denotes the ball centered at the origin with radius $R$. Please notice that $\sigma_{r}$ is a first hitting time of a one-dimensional Brownian motion.
Let us fix $r>0$ and $n \in \mathbb{N}$. Then, we find that \begin{align*} &P(\tau_{nr^{1/2}} \le \sigma_{r})=P(r^{-1} \tau_{nr^{1/2}} \le r^{-1} \sigma_r)\\ &=P(\tau_{n} \le r^{-1} \sigma_r)=P(\tau_n \le \sigma_{r^{1/2}}). \end{align*} In the above display, we used the scaling property of Brownian motion, which implies that $r^{-1} \tau_{nr^{1/2}}=\tau_{n}$ and $r^{-1} \sigma_r=\sigma_{r^{1/2}}$ in the sense of distribution. Is my argument correct?