$ \hat{F}_n(b) := \frac{1}{n} \cdot$ |{ $i \in$ {$1,...,n$} : $x_i \leq b$}| $ =\frac{1}{n} \sum_{i=1}^{n} \mathbb{1}_{x_i}((-\infty,b]). $ .
We want that:
$\hat{F}_n(b)$ is non-decreasing. (Check)
$\hat{F}_n(b)$ $\rightarrow 0$ for $b \rightarrow -\infty$.
$\hat{F}_n(b)$ $\rightarrow 1$ for $b \rightarrow \infty$.
$\hat{F}_n$ is right-continuous, which means $\hat{F}_n(b_{-}) = \lim_{c \nearrow b}\hat{F}_n(c) $.
I hope that I didn't forget one criterion. Here is my idea for the properties: Since $\mathbb{1}_{x_i}((-\infty,-\infty)) = 0$ and $\mathbb{1}_{x_i}((-\infty,\infty)) = 1$ we have that $\hat{F}_n(b)$ $\rightarrow 0$ for $b \rightarrow -\infty$ and $\hat{F}_n(b)$ $\rightarrow 1$ for $b \rightarrow \infty$. Is this enough? How can I show that $\hat{F}_n$ right-continuous?