  We use the tridiagonal matrix representation to derive a local semicircle law
for Gaussian beta ensembles at the optimal level of $n^{-1+\delta}$ for any
$\delta > 0$. Using a resolvent expansion, we first derive a semicircle law at
the intermediate level of $n^{-1/2+\delta}$; then an induction argument allows
us to reach the optimal level. This result was obtained in a different setting,
using different methods, by Bourgade, Erd\"os, and Yau and in Bao and Su. Our
approach is new and could be extended to other tridiagonal models.
