I understand that I have been offensive in my actions the last time that we were together. I do not mean to be so rude and the ill feelings that I might have caused were unintentional.
I am writing to convey to you my sincere apology for what has happened. If there is anything that I can do for you to forgive my misgiving, please do not hesitate to let me know. I want you to know how sorry I am upon learning of my mistake. From the years that I have known you, I am not sure how many times I might have made a mistake to you.
Please accept the apology that I am offering you. I promise that next time, I will be more prudent in my actions and words. I hope you understand.
Thank you very much and I hope we can always be friends.