# Convergence in Probability of i.i.d exponential random variables

I'm a bit lost here, the question goes as follows:

Suppose that $X_k$ are i.i.d. and follow an exponential distribution with parameter $\lambda$.

Define $F_n(x) := \frac{1}{n}\sum_{i=1}^n(X_{k}\leq x)$ for x $\geq0$

Question :Show that $F_n(x)$ converges in probability to $1−e ^{−\lambda x}$ Does it also converge in $L^{1}$ norm?

I think to show that $F_n(x)$ converges in probability we need to show that:

$\lim_{n\to\infty}P\left(\left|F_n(x)-F(x)\right|>\varepsilon\right)=0$

Therefore we can state that: