Prove $\lim |u(x,y)|e^{-|x|}=0$ for harmonic function $u$

Question

Let $\epsilon >0$ and $\Omega \subset \{(x,y): -\frac{\pi}{2}+\epsilon<y<\frac{\pi}{2}-\epsilon\}$. Let $u$ be harmonic function in $\Omega$ such that $u=0$ on $\partial \Omega$. Prove that if $\Omega$ is unbounded and $$\lim_{(x,y)\in \Omega,|x|\to \infty}|u(x,y)|e^{-|x|}=0$$ then $u\equiv 0$ in $\Omega$.