criterion for borel sets
i am trying to show that for a function f:X->R to be borel-measurable, it is sufficient that one of the following conditions is met:
1. f^(-1)(]a,b[) is in A for all a,b in R, a<b where A is the set of measurable sets.
2. f^(-1)([a,b]) is in A
3. f^(-1)(]a,+infinite[) is in A
4. f^(-1)(]-infinite,a[) is in A
5. f^(-1)([a,+infinite[) is in A
6. f^(-1)(]-infinite,a]) is in A
i know that if i can show 1 then it shouldnt be hard to show 1 implies 2, 2 implies 3, etc. since i already showed that each set can be generated by complements, unions, and countable intersections of other sets, but i can't seem to show that condition 1 is true. some help please? thanks.
For condition 1, use the fact that every open subset of R is a countable union of open intervals.