Suppose that \(\left\{F_{n}ight\}_{n=1}^{\infty}\) is a sequence of distribution functions such that

\[\lim _{n ightarrow \infty} F_{n}(x)=F(x)\]

for all \(x \in \mathbb{R}\) for some function \(F(x)\). Prove the following properties of \(F(x)\).

a. \(F(x) \in[0,1]\) for all \(x \in \mathbb{R}\).

b. \(F(x)\) is a non-decreasing function.