The phrase "persists in being" is correct and usable in written English.
It is often used to describe someone or something that continues to stay the same, despite any changes or difficulties. For example: "Despite the many challenges he has faced, Jack persists in being a positive role model for his children."
One page further along, she persists in being "crude and illiterate," and is now "slovenly".
Though he can't afford Rosalind, he persists in being heartbroken by her -- a sign that his larger ambition is unsurrendered.
And while my attitude used to be if you don't trust them, don't sleep with them, the world persists in being more complicated and darker than that.
While it's common knowledge that the cabinet should be routinely sorted through, with out-of-date pills and other medications thrown out, the cabinet persists in being an archive of old afflictions.
Why is it then that the "question of animal culture" persists in being a controversial, unsolved puzzle?
Disproving the null model, sophistication persists in being significant even accounting for the shop's volume and variety.
Francis James Child said that Lowell, whom he deemed was typically "perverse," was able to "persist in being serious contrary to his impulses and his talents".
This is a monumentally naff film, shaming and depressing in a way that British feature-film comedies have persisted in being, intermittently, all our lives.
If the Hurricanes are to persist in being rude guests, they will need more strong work from their highly underrated goalie, Cam Ward.
If Japanese markets persist in being spectacularly inefficient, then investors will move their business to places where taxes and other transaction costs are lower.
O'Reilly delights in presenting himself as a nonpartisan truth-teller, giving viewers the straight dope; if he tends to heap outrage on liberals, the story line goes, that is because they persist in being especially outrageous.
