"How to verify that ethanol is infinitely soluble in water?".
You cannot do it because it is WRONG. And not because it is false for ethanol, it is simply plain wrong conceptually and mathematically.
Solubility is the ratio between the amount of the specified substance and the volume of the phase solution.
As for a substance occupies its own volume, solubility cannot tend to infinity but at most reach the value dictated by its molar volume in the given P, T conditions.
For instance, let me take water - for which I remember the value - instead of ethanol. Its concentration as pure substance is about 55.5 mol/l.
This is aimed to correct a disturbing error in the text book rather than provide a practical answer.
For finding out if water and ethanol are miscible in any ratio - this should have been the question - the book is at least correct that a simple test tube suffices to get at least a reasonable idea. How and why is already in the other answers and comments and I assume it is clear to the OP as well, at this point.