What were the clues that Holden understood it was impossible to clean all the mess of the world ?In the ending parts mostly.
Holden originally fantasizes as being the 'Catcher in the Rye,' the person who could prevent children from falling off of a metaphorical cliff, who could save them from the eventual disillusionment of adulthood. This viewpoint was how Holden was going to fix all the mess in the world. Gradually, Holden begins to change his perspective; by the end of the novel, Holden has an epiphany as he watches his sister Phoebe ride the carousel and try for the gold ring. Holden comments:
"The thing with kids is, if they want to grab for the gold ring, you have to let them do it, and not say anything. If they fall off, they fall off, but it's bad if you say anything."
Holden's new insight is the direct opposite of his original view of being the 'catcher;' Holden now sees that part of letting children grow up is that they may "fall off" or miss that gold ring. The difference is that now Holden is at peace with the possibility of failure. "If they fall off, they fall off." In this moment, Holden experiences true happiness, watching Phoebe and standing in the rain.