Let $S_n$ be a simple symmetric random walk on the square lattice $\mathbb{Z}^2$ with $S_0=(0,0)$. That is, the walker starts from the origin and at each step independently, she steps one unit to East, North, West or South with equal chance. Denote by $D_n$ the walker's Euclidean distance from the origin of $\mathbb{Z}^2$ at time $n$, and let $\nu_r=\inf \left\{n: D_n>r\right\}$. (a) Show that $D_n^2-n$ is a martingale. (b) Show that $r^{-2} \mathbb{E} \nu_r \rightarrow 1$ as $r \rightarrow \infty$.
Could someone verify my answer for part b)
$$\mathbb{E}\left[D_{\nu_r}^2-\nu_r\right]=0$$ $$\mathbb{E}\left[\nu_r\right]=\mathbb{E}\left[D_{\nu_r}^2\right] $$ $$\mathbb{E}\left[D_{\nu_r}^2\right]=\mathbb{E}\left[r^2\right] = r^2$$ $$\frac{\mathbb{E}\left[\nu_r\right]}{r^2}=1$$ $$r^{-2} \mathbb{E}\left[\nu_r\right] \rightarrow 1$$
am I correct in assuming $\mathbb{E}\left[D_{\nu_r}^2\right]=\mathbb{E}\left[r^2\right] = r^2\\$?