Constant Hazard Rate from Memorylessness

A device's lifetime $X$ has survival function $\bar{F}(t) = P(X > t)$ and hazard rate $h(t) = f(t)/\bar{F}(t)$. Show that the memoryless property $P(X > s + t \mid X > s) = P(X > t)$ implies $h(t) = \lambda$ (a constant) for all $t \geq 0$, and conversely that a constant hazard rate implies the memoryless property. Conclude that $X \sim \operatorname{Exp}(\lambda)$.