We do not give things up for the sake of giving things up. We give things up for the sake of being closer to God and obeying Him more fully. The things we give up are of no value to God. What is of value to God is my very life.
We do not give things up because doing so brings us closer to God. We give things up because we finally recognize nothing else is more important than being closer to God. We give them up "for the sake of the only thing worth having". Getting rid of things does not, themselves, bring us closer to God. Getting rid of things loosens our grip on the things which keep us from grasping a stronger relationship with God.