To remain above the exchange limit for $k$ steps requires that the
current ideal price exceeds by $k\ln \kappa_u$ the exchange price
$k$ days earlier. Since,
\[
   Y = a + Z_1+\ldots +Z_k \sim {\cal N} ( a + k\mu, k\sigma^2),
\]
then,
\begin{eqnarray*}
      \Prob(\, Y>k\ln \kappa_u \,) &=& {1 \over \sqrt{2\pi k \sigma^2 }}
      \int_{k\ln \kappa_u}^{\infty}
       e^{-(x-k\mu-a)^2/(2k\sigma^2)}\,dx
       \\
       &=& {1\over \sqrt{2 \pi}} \int_{(\lambda\sqrt{k})-a/(\sigma\sqrt{k})}^{\infty}
       e^{-x^2/2}\,dx.
\end{eqnarray*}
Because $\lambda>0$, we can select a $K$ such that for all $k\ge
K$,
\[
  (\lambda\sqrt{k})-a/(\sigma\sqrt{k})> \lambda\sqrt{k/2} > 0
\]
and we can use the approximation for the tail of the normal
distribution,
\[
    {1\over\sqrt{2 \pi}}\int_{x}^\infty e^{-y^2/2}\, dy <
    {1\over x \sqrt{2 \pi}}\, e^{-x^2/2}
\]
for $x>0$, see Feller, Vol. 1, Lemma VII.1.7. Since the bound is
increasing with decreasing $x$, for simplicity of notation we set
$x=\lambda\sqrt{k/2}$ to complete the proof.