Proving sets are Borel sets

Question

Suppose $f,g: \mathbb{R}^n \rightarrow [0, +\infty]$ are Borel measurable functions. Consider the sets $A = \{x \in \mathbb{R}^n | f(x) < g(x)\}$ and $B = \{x \in \mathbb{R}^n | f(x) = g(x)\}$. Are either of these sets Borel sets?

I'm still quite new to Borel-$\sigma$-algebras. I thought of proving that these sets were either open or closed or could be written as an interval, but I wasn't able to get really far. Any hints are welcome!

Answer 1

Hint

1) $$\{x\mid f(x)<g(x)\}=\bigcup_{r\in \mathbb Q}\{x\mid f(x)<r\}\cap \{x\mid g(x)>r\}.$$

2. $$\{x\mid f(x)=g(x)\}=\{x\mid f(x)<g(x)\}^c\cup\{x\mid g(x)<f(x)\}^c.$$