Reading the Robbins–Monro conditions off the proof

From the recursion $e_{t+1}\le(1-\eta_t\mu)e_t+\eta_t^2\sigma^2$, explain why the conditions

$$\sum_t\eta_t = \infty,\qquad \sum_t\eta_t^2<\infty$$

are exactly what is needed for $e_t\to0$. Check which of $\eta_t = \eta$, $\eta_t = c/\sqrt t$, $\eta_t = c/t$ satisfy them, and what each gives.