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No it wouldn't be inappropriate to send her a letter telling her exactly how you feel about her turning her back on you and just forgetting about the friendship for another friendship. Make sure that the letter covers every thing you want her to know and also tell her you aren't really looking for a friendship with her because friends wouldn't teach other friends that way, but inform her that if she wants to keep good friends in her life maybe she shouldn't leave friends that have been there for her for friends she hardly knows. Make sure that when you send this letter to her that you would have said everything you wanted to say or maybe she didn't give you a chance to say. If this is the only way you feel you can move on then don't hesitate to write her the letter and send it as soon as you are done and proof read it to make sure it says everything you wanted to say.