I took it to mean that the problem would not be given in this way unless there was an exact solution. In the stated version, there are no degrees of freedom specified, so there can be no degrees of freedom in the solution. In that way, the proof wouldn't work if the problem stated "A six inch high cylindrical hole is drilled through the center of a sphere with radius R. How much volume is left in the sphere?"

In that case, the solution might be dependent on R, and we'd have to go through the long version. Without that mention of R, unless the problem is wrongly stated, it must be constant.

So it's still an invalid inductive proof, but it's a lot stronger than just assuming that because there is a question there is a constant answer.