# Show that every interval is a Borel set

Show that every interval is a Borel set.

My textbook states:

The intersection of all the $\sigma$-algebras of subsets of $\mathbb{R}$ that contain the open sets is a $\sigma$-algebra called the Borel $\sigma$-algebra; members of this collection are called Borel sets.

Since $(-\infty, a)$, $(a,b)$, $(a,\infty)$ are open, they must be Borel. For any interval of the following forms, we also see that they can be represented by open intervals.

\begin{align} [a,b]&=\bigcap^{\infty}_{n=1} (a-1/n, b+1/n)\\ (a,b]&=\bigcap^{\infty}_{n=1} (a, b+1/n)\\ [a,b)&=\bigcap^{\infty}_{n=1} (a-1/n, b) \end{align}

Do you think my answer is correct?

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Yes, it looks fine. –  Brian M. Scott Feb 6 '13 at 9:43
I agree with @Brian M. Scott. You can convince yourself that equality holds by showing the RHS is a subset of the LHS and vice versa. Let us verify that $$[a,b]=\bigcap_{n=1}^\infty (a-\tfrac{1}{n},b+\tfrac{1}{n})$$ is indeed true.
Therefore, take an $x\in [a,b]$, i.e. $a\leq x\leq b$. Then since $\tfrac{1}{n}>0$ for all $n\in\mathbb{N}$ we have that $$a-\frac{1}{n}<x<b+\frac{1}{n},\quad\text{for all }\,n\in\mathbb N,$$ and hence $x\in \bigcap\limits_{n=1}^\infty (a-\tfrac{1}{n},b+\tfrac{1}{n})$.
Now, take an $x\in \bigcap\limits_{n=1}^\infty (a-\tfrac{1}{n},b+\tfrac{1}{n})$, i.e. $$a-\frac{1}{n}<x<b+\frac{1}{n},\quad\text{for all }\,n\in\mathbb N.$$ Assume for contradiction that $x\notin [a,b]$. Without loss of generality we can assume that $x>b$. Now, there exist an $n\in\mathbb{N}$ (pick $n$ such that $\tfrac{1}{n}<|x-b|$) such that $$x> b+\frac{1}{n}>b$$ which is a contradiction.