come round to (someone's) way of thinking

Translations
прийти к (чьей-то) точке зрения
After a long discussion, she finally came round to his way of thinking.
После долгого обсуждения она наконец-то пришла к его точке зрения.
Definitions
To change your opinion or perspective so that it matches someone else's; to be persuaded to agree with another person's viewpoint.