  This paper proposes an adaptive timestep construction for an Euler-Maruyama
approximation of the ergodic SDEs with a drift which is not globally Lipschitz
over an infinite time interval. If the timestep is bounded appropriately, we
show not only the stability of the numerical solution and the standard strong
convergence order, but also that the bound for moments and strong error of the
numerical solution are uniform in T, which allow us to introduce the adaptive
multilevel Monte Carlo. Numerical experiments support our analysis.
