"我写信给你的目的是" 可以用英语表示为 "The purpose of me writing this letter to you is..."。
其中 "purpose" 是名词，表示 "目的"，而 "writing" 是动词，表示 "写"。
例句：
The purpose of me writing this letter to you is to apologize for my mistake. (我写这封信给你的目的是为我的错误道歉。)
The purpose of me writing this email is to inform you of a change in the meeting schedule. (我写这封邮件的目的是通知你会议时间的更改。)
The purpose of me writing this memo is to remind you of the deadpne for the project. (我写这份备忘录的目的是提醒你项目的截止日期。)