Limit distribution for a random walk with absorption

Let $\xi_1,\xi_2,\dots$ are independent identically distributed random variables, $\mathbf M\xi_1=0$, $\mathbf D\xi_1=1$ and
$$ S_n=n^{-1/2}(\xi_1+\dots+\xi_n),\qquad\nu=\min\{n:S_n<0\}. $$
We show that
$$ \mathbf P\{S_\nu<x\mid\nu>n\}\to V(x),\qquad\mathbf P\{S_n<x\mid\nu>n\}\to 1-e^{-x^2/2}. $$