Delay differential equations

Consider the differential-delay equation given by $y'(t) = -y(t-t_0)$. Here, the independent variable $t$ is a real variable, the function $y$ is allowed to be complex valued, and $t_0$ is a positive constant. Prove that if $0 < t_0 < \pi /2$ then every solution of the form $y(t) = e^{\lambda t}$, with $\lambda$ complex, tends to $0$ as $t \to +\infty$.