How to prove that ECDF is right-continuous?

Question

Let $X_i$ iid. How to prove that the function $F_n(x,\omega)=\dfrac{1}{n}\sum\limits_{i=1}^n\textbf{1}_{(-\infty,x]}(X_i(\omega))$ for a given $n$ and $\omega$ is right-continuous?

Answer 1

$1_{(-\infty, x]} (y)$ is right-continuous for any $y$. [ Note that $1_{(-\infty, x]} (y)=1$ if $x \geq y$ and $0$ if $x <y$]. Linear combinations of right-continuous functions is right-continuous.