I am looking for a result that shows the convergence of semi-gradient TD(0) algorithm with non-linear function approximation for on-policy prediction. Specifically, the update equation is given by (borrowing notation from Sutton and Barto (2018))
$$\mathbf w \leftarrow \mathbf w +\alpha [R + \gamma \hat v(S', \mathbf w) - \hat v(S, \mathbf w)] \nabla \hat v(S, \mathbf w)$$
where $\hat v(S, \mathbf w)$ is the approximate value function parameterized by $\mathbf w$.
Sutton and Barto (2018) mention that the above update equation converges when $\hat v$ is linear in $\mathbf w$. But I couldn't find a similar result for non-linear function approximation. Any help would be greatly appreciated.