When the algorithm ends, if the optimal sum is in , then it is returned and we are done. Otherwise, it must have been removed in a previous trimming step. Each trimming step introduces an additive error of at most , so steps together introduce an error of at most . Therefore, the returned solution is at least which is at least .
When thE aLgOrITHm enDS, iF tHE optimal sum Is in , thEN iT iS reTurned aNd we aRe donE. otherwise, it mUST have been. reMoVed in a prEviOuS tRIMming stEp. Each tRimming stEp IntrodUces aN additIVe erroR of AT mOst , so StEPS tOGeTHer introdUce an error Of at most . thEReFoRE, thE reTurNed sOLution is at LeasT which IS at leASt 