Prove that the union of increasing $\sigma$-field is not a $\sigma$-field

Find examples where $\mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots$ are increasing $\sigma$-fields but $\bigcup_{n = 1} ^\infty \mathcal{F}_n$ is not. I have constructed an example that I believe is correct, but having trouble to show this is actually the right counter-example:

Define $\mathcal{F}_n = \sigma(\{ [0, i]: i \leq n \})$ on $\mathbb{R}$. Then I believe $\bigcup_{i = 1} ^\infty [0, i] = [0, \infty) \not\in \bigcup_{n = 1} ^\infty \mathcal{F}_n$.

However, I am having trouble to show $[0, \infty) \not\in \bigcup_{n= 1} ^\infty \mathcal{F}_n$. Any suggestions?