I am reading the paper *Nodal sets of eigenfunctions of the Laplacian on Surfaces* by Donnelly and Fefferman available here. I have a problem understanding Lemma 5.10. To my understanding, what follows is the content of the Lemma 5.10

Let $\mathbb{D}$ be the unit disc in $\mathbb{R}^2$ centered at the origin. Fix $m \in \mathbb{N}$. Then there is a constant $C_m >0$ such that for any smooth $g :\mathbb{D} \rightarrow \mathbb{R}$ satisfying the following condition \begin{align*} \max_{|\alpha| \leq m} \left| \frac{\partial^{\alpha}g}{\partial z^{\alpha}}(0) \right| \geq 1 \end{align*} there are a pair of perpendicular directions $v_1$ and $v_2$ and some $k \leq m$ (here $v_1, v_2$ and $k$ can possibly depend on the function $g$) such that the following holds \begin{align*} |\partial_{v_1}^{k}g(0)| \geq C_m~\mbox{and}~|\partial_{v_2}^{k}g(0)| \geq C_m \end{align*}

Here are my questions regarding this Lemma.

Firstly, is my understanding of the statement of Lemma 5.10 correct? More specifically, is it claimed that there is a universal constant $C_m$ (independent of the function $g$)?

If the answer to my first question is Yes, I cannot really see how the proof provided in the paper establishes existence of the universal constant $C_m$.

Thanks in advance!