Yes, it generally means "to do something in vain". In the context of this sentence "J'ai beau me dire qu'au fond c'est mieux", I would translate it with this meaning "I'm trying to persuade myself it's better this way in the end". Here are some more examples : J'ai beau travailler dur, mon patron n'est jamais satisfait. (though I'm working hard, my boss is never satisfied) J'ai beau aller au lit à 21 heures, je n'arrive jamais à me réveiller le matin (even if I go to bed at 9pm, i never get to wake up on time in the morning). There's the notion that even if you're trying your best to do something, you never get the results that you hope for. I hope it's clearer now. If you have more questions, don't hesitate to conatct me! À bientôt et bon courage dans tes études du français!