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The new version of the gedanken experiment proposed by Sorce and Wald hasbeen used to examine the weak cosmic censorship conjecture (WCCC) for blackholes at the second-order approximation of the matter fields perturbation.However, only considering the perturbation until the second-order approximationis incomplete because there is an optimal option such that the existingcondition of the event horizon vanishes at second-order. For this circumstance,we cannot judge whether the WCCC is satisfied at this order. In ourinvestigation, the $k$th-order perturbation inequality is generally derived.Using the inequalities, we examine the WCCC for nearly extremalReissner-Nordst\"{o}m black holes at higher-order approximation. It is shownthat the WCCC cannot be violated yet after the perturbation. From this result,it can be indicated that the WCCC is strictly satisfied at the perturbationlevel for nearly extremal RN black holes.

Gedanken experiments at high-order approximation: nearly extremal Reissner-Nordstrom black holes cannot be overcharged