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## Homework Statement

Suppose [itex]\Omega = \mathbb{R}[/itex]. Let [itex]A_n = [0,\frac{1}{n}][/itex]. Let [itex]\mathcal{A}=\{A_n:n\in\mathbb{N}\}[/itex].

Is [itex]\{0\}[/itex] an element of [itex]\mathcal{F}=\sigma(\mathcal{A})[/itex]?

## Homework Equations

## The Attempt at a Solution

Clearly [itex]\lim_{n\rightarrow\infty} A_n = \{0\}[/itex]. Not sure how to show that it need be in [itex]\mathcal{F}[/itex] though.