In part (ii), you showed that
so long as a, b, and c are real and positive.
where x, y, and z are suitably chosen positive reals. Putting these into the equation from part (ii):
and thus you get the solution's inequation (1), except in x, y, and z.
Similar substitutions yield the other three inequations, and they sum to give the result required.
The problem with this approach is that the substitutions collectively require a, b, and c to be less than 1.