We conclude via proof by contradiction that $j_b(x)$ has no zeros.

[T2.5]{} For each $b$ and each band $j$, there is an integer $D_{b,j}$ so the number of zeros $N_{b,j}(m)$ of $p_{mp+b-1}$ is either $m-D_{b,j}$ or $m-D_{b,j}+1$. In particular, $${\label}{2.49} 
\sup_{n,j}\, \biggl| {\frac}{n}{p} - N^{(n,j)}\biggr| < \infty$$ Moreover, holds.

By , , , and , we have $${\label}{2.50} 
p_{mp+b-1}(x) = j_b(x) \Gamma_+(x)^m + {\overline}{j_b(x)}\,\, {\overline}{\Gamma_+(x)}^m$$ on the bands. By the lemma, $j_b(x)$ is nonvanishing inside band $j$, so $${\label}{2.51} 
j_b(x) = {\lvertj_b(x)\rvert} e^{i\gamma_b(x)}$$ where $\gamma_b$ is continuous — indeed, real analytic — and by a simple argument, $\gamma_b$ and $\gamma'_b$ have limits as $x\downarrow a_j$ or $x\uparrow b_j$.

By , becomes $${\label}{2.52} 
p_{mp+b-1}(x) = 2{\lvertj_b(x)\rvert} \cos (\pi mp\, k(x) + \gamma_b(x))$$ Define $D_{b,j}$ to be the negative of the integral part of $[\gamma_b(b_j)- 
\gamma_b(a_j)]/\pi$. Since $\sup_{\text{bands}} {\lvert\gamma'_b(x)\rvert}<\infty$, there is, for large $m$, at most one solution of $\pi mp\, k(x) + \gamma_b(x) =\pi\ell$ for each $\ell$. Given this, it is immediate that the number of zeros is $m-D_{b,j}$ or $m-D_{b,j}+1$.

Finally, is immediate from . Given that $\gamma$ is $C^1$, we even get that $${\label}{2.53} 
k(x_{\ell+1}^{(n,j)}) - k(x_\ell^{(n,j)}) = {\frac}{1}{n} + O\biggl( {\frac}{1}{n^2}\biggr)$$

As for point (3) from the introduction, 