  The strong convergence of numerical methods for stochastic differential
equations (SDEs) for $t\in[0,\infty)$ is proved. The result is applicable to
any one-step numerical methods with Markov property that have the finite time
strong convergence and the uniformly bounded moment. In addition, the
convergence of the numerical stationary distribution to the underlying one can
be derived from this result. To demonstrate the application of this result, the
strong convergence in the infinite horizon of the backward Euler-Maruyama
method in the $L^p$ sense for some small $p\in (0,1)$ is proved for SDEs with
super-linear coefficients, which is also a a standalone new result. Numerical
simulations are provided to illustrate the theoretical results.
