Let the real process \((X_t)\) solve a stochastic differential equation \(({\mathcal E})\) which is driven by a real Brownian motion \((B_t)\) and has nice coefficients. Let \((X^n_t)\) be the solution of some difference equation associated with \(({\mathcal E})\), \((B_t)\) being replaced by smooth approximation. Then it is proved that \(X^n_t\to X_t\) in \(L^2\), uniformly for \(0\leq t\leq 1\), under some assumption on the mesh of the discretization.