Dear [Name],
I am writing to apologize for not believing in your ability or goals. I know that this lack of faith has caused you pain and I am deeply sorry for that.
I understand that you have worked hard to achieve your goals and that my lack of belief has been a hindrance to your progress. I should have had more faith in your abilities and I regret not doing so.
I want you to know that I recognize the effort you have put into your work and I am proud of the progress you have made. You have shown me that you are capable of achieving great things and I am sorry for not recognizing that sooner.
I know that I cannot take back my words, but I am committed to being a better supporter for you in the future. I will do my best to be more encouraging and to have faith in your abilities.
I hope that you can forgive me for my lack of belief in you. I am sorry for not being the kind of friend that you deserve.
Sincerely,
[Your Name]