Can satellites moving in orbits that lie in different planes remain at the same distance from each other at all times? How can I prove that they cannot?
They can in at least once scenario, thus proving it false isn't possible.
One orbit circular, the other elliptical, at 90° to each other.
When the polar orbiting satellite crosses the equator have the equitorial orbiting satellite positioned opposite (so they don't collide). When the polar orbiting satellite is passing over a pole have the equitorial orbiting satellite at 90° from where it was, as described in the prior sentence. An elliptical orbit maintains the distance.
It is possible for more than two satellites to do this simultaneously, so there's more than one solution.