Can we say renunciation means giving up?

Swami Vivekananda said” We have to cover everything with the Lord Himself, not by a false sort of optimism, not by blinding our eyes to the evil, but by really seeing God in everything. Thus we have to give up the world, and when the world is given up, what remains? God. What is meant? You can have your wife; it does not mean that you are to abandon her, but that you are to see God in the wife. Give up your children; what does that mean? To turn them out of doors, as some human brutes do in every country? Certainly not. That is diabolism; it is not religion. But see God in your children. So, in everything. In life and in death, in happiness and in misery, the Lord is equally present. The whole world is full of the Lord. Open your eyes and see Him. This is what Vedanta teaches”.
So long as you have body there will be world. “I” the egotism and the world is not different from one another. If you say I then you are not different from the world and if you see the world then there should be an I to see it, so you cannot renounce the world alone. To give up the world you have to give up the I consciousness. Giving up I forever is known as renunciation as per my understanding, what ever you give up apart from this, cant be real renunciation as long as there is I thought. Every thought starts and end in I.. so what is the use of renouncing everything if you cant give Up I or me and mine? And i don't believe there are two sort of renunciation for monks and others. Way is only one people call it in different names.