We have to keep in mind that the Taming of the Shrew is a play within a play, meaning that Shakespeare is going to pains to actually show us the unreality of a man actually being able to successfully "tame" his wife as Petruchio does.
That being said, this is an easy thesis to prove because the text provides plenty of evidence that Kate has become a docile wife by the end of the action. One way Petruchio tames her is through killing her with "kindness": being so over solicitous about food and drink that Kate can neither eat nor sleep. He also whisks her away from her wedding very quickly and isolates her away from friends and family. With no one to turn to, not enough to eat, and no sleep, three main methods Petruchio uses to control her, she capitulates. What he does is sadistic, but apparently—at least in this fictive universe—effective.
Proof of Kate's new role as obedient wife can be found in the following quote from act 5:
Thy husband is thy lord, thy life, thy keeper,
Thy head, thy...
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