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In this paper, we prove the Melnikov's persistency theorem by combining thetraditional Kolmogorov-Arnold-Moser (KAM) technique and theCraig-Wayne-Bourgain (CWB) method. The aim of this paper is twofold. One is toestablish the linear stability of the perturbed invariant tori by using the CWBmethod without the second Melnikov condition. The other one is to illustratethe CWB method in detail and make the CWB method more accessible.

On linear stability of KAM tori via the Craig-Wayne-Bourgain method