Reduce[ForAll[x, 0 < x < [Pi]/2, 2 a Sin[a x] >= 0], a, Reals]
Probably the Reduce function (and other functions that use quantifiers, like ForAll and Exists) are based on an algorithm called Cylindrical Algebraic Decomposition (CAD). This algorithm is absolutely brilliant, but it has one huge drawback: it only works with polynomials.
Removing ForAll from inside Reduce leads to a clear answer:
Reduce[0 < x < \[Pi]/2 &&
2 a Sin[a x] >= 0 &&
0 <= a x <= 2 Pi,
{a, x}]
You can reintroduce ForAll to scan the answer:
red = Reduce[0 < x < \[Pi]/2 &&
2 a Sin[a x] >= 0 &&
0 <= a x <= 2 Pi,
{a, x}]
Map[Reduce[ForAll[x, 0 < x < \[Pi]/2, #]] &,
List @@ red]
Thanks for your reply. Your answer gets to the heart of the issue, explaining the underlying algorithm and its scope of application in the code implementation. The equivalent expansion of the last line of code gives the final desired result
Thanks for your reply.