Proof that exists $L^2$ limit $$ \lim_{\varepsilon\downarrow 0} L(t,\varepsilon)=\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon}\int_0^t\mathbf{1}\left(W_s\in(-\varepsilon,\varepsilon)\right)ds, $$ where $W_s$ is a Brownion motion and $\varepsilon>0$ and $t$ is a fixed number.
I really don't know how to proof this. Any help would be appreciated. Thanks!