Consider a SDE $dX_t=f(X_t)dt+\sigma(X_t)dW_t$ $(t\in [0,T], T<\infty)$ where $W$ is a Wiener Process.
Let $X$ be its solution and $Y^{\Delta_N}$ ($N \in \mathbb{N})$ its approximation by Euler-Maruyama, where $\Delta_N$ denotes the step-width, i.e. for $\{ 0=t_0<t_1<...<t_m=T \}$ ,$\Delta_N=t_i-t_{i-1}$ (which is constant).
From the lecture I know that $ E|Y^{\Delta_N}_T-X_T|\leq C \Delta^{\frac{1}{2}}_N$ for all $\Delta_N$ and some constant C
Now suppose $\Delta_N \leq C_2N^{-\alpha}$ for some $C_2$ and $\alpha >1$.
I have to show that $Y^{\Delta_N}_T \to X_T$ almost surely.
I tried using Borel-Cantelli but it only works if $\alpha > 2$. How can i proof it for $\alpha>1$?