Suppose I have $X_1, X_2,...$ are iid with $Exp(\lambda)$ with $$G_{\lambda}(x)=1-exp(-\lambda x), \ \ \ \ \ x>0$$ Set $$G_n(x)=1-exp(-\hat{\lambda}_n x), \ \ \ \ \ x>0$$ with $$\hat{\lambda}_n=\frac{n}{\sum_{i=1}^{n}X_i} $$ I would like to show that $$\sup_x |G_n(x)-G_{\lambda}(x)| \rightarrow 0 \ \ \ \ a.s$$
I have tried to use the mean value theorem, for example, $$|G_n(x)-G_{\lambda}(x)| = |exp(-\lambda x) - exp(\hat{\lambda}_n x)| \\ = |xexp(-\lambda'x)| |\hat{\lambda}_n - \lambda| \\ <x|\hat{\lambda}_n - \lambda| $$
I am not sure whether this is the correct move and hope someone would give me some suggestions.