Every continuous function is Borel measurable.

Question

We say that $f$ is Botel measurable function if inverse image of every Borel set is again is Borel measurable.

Equivalent: $f: X\rightarrow R$ ($X\subset R$) is borel measurable set if $f^{-1}(a,\infty)$ for any $a$ is Borel measurable set.

Suppose $f:X\rightarrow R$ is continuous function. $X\subset R$. Prove that this function is Borel measurable.

1. We know that if a function is continuous then the inverse image of every open set is open.

2. Hence $f^{-1}(a,\infty)$ will be open for every $a$ in $R$.

3. Now we have that inverse image of $(a,\infty)$ (which is Borel set because it is open) is a Borel set.

4. $f$ is Borel measurable function.

Question: Am I correct? There is very different proof in my textbook. My proof is very easy. So I am sure there is big mistake in my proof. What is wrong?

EDIT: