Show that every interval is a Borel set

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.

We have shown the two inclusions and conclude that the sets are the same.

You could have tried it easier. For example, Finte sets are are closed in R . The compliments of which are union of open sets, are borel and hence any finite set is borel. You can write a clopen or a closed interval as union of an open interval and a finite set and so they are borel!