Suppose I have a Brownian motion $X_t$ in the interval $[0, \gamma]$ which is reflected at the boundaries $0$ and $\gamma$. We call $P_\gamma$ the law of such reflected Brownian motion. The generator is defined on continuous functions $f:[0, \gamma]\to\mathbb R$ such that $f'(0)=f'(\gamma)=0$ in the following way
$$L_\gamma f(X)=\frac{1}{2}f''(X)$$ I would like to prove that
$$\lim_{\gamma\to 0}\mathbb E_{P_\gamma}(f''(X))=0.$$
Has anyone any idea?