Following this definition: A subset $X$ of $\mathbb{R}$ is said to be open if for every real number $x \in X$ , there is an open interval $(a,b)$ with $x\in(a,b)$ and $(a,b)\subset X$.
That's how I tried:
Take $x \in (-\infty,b)$. Then it follows that $x<b$ . Since $(-\infty,b)$ is not bounded below, there is $a\in (-\infty,b)$ such that $a<x$. Then $x \in (a,b)$. Also $(a,b) \subset (-\infty,b)$. Therefore $(-\infty,b)$ is an open set in $\mathbb{R}$.
Is this correct ?