Prove that a set is a Borel set

Question

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we know that f and g are two Borel functions. We want to prove that $\{x:f(x)<g(x)\}$ is a Borel set.

I know that for a set A, the inverse of f(A) and g(A) will be a Borel set. I don't see how i can prove the other thing with this information.

Answer 1

$h=f-g$ is a Borel function and $\{x: f(x) <g(x)\}=h^{-1} (-\infty ,0)$.

Answer 2

HINT

Prove that $$\{f<g\}=\bigcup_{q \in \Bbb{Q}}(\{f<q\}\cap \{g>q\})$$