Is it a good idea to send a hand written apology letter to an ex girlfreind that you hurt?

I've been thinking about this long and hard for a long time now and I keep going back and forth on this.

I really want to send a letter to my ex girlfriend whom I love very much abd miss a lot but ended up losing her due to cheating. It's been a year since we broke up.

I want to send her a letter apologizing for everything that I did wrong. Why I did what I did. What caused me to do what I did. To show how her how much I've changed in my life in a positive way. How I've progressed my life in school, in my career and financially. How I've gotten sober and quit drinking. How I've matured as a person and learned and grown from my mistakes.

I also want to tell her that I understand why she left, how she had to and that she had no choice. I betrayed her trust and hurt her. I want to show her that I understand the damage that I caused. I want her to know that i take responsibility for damaging/ruining our relationship. I want her to know that I don't blame her and I dont fault her. That I blame myself. That I was selfish.

I also want to tell her from the bottom of my heart how I really feel about her. How much she really means to me. And how serious i took this breakup to change my life for the better. I want tell her that I love her with all my heart.

I also want to let her know that I dont expect her forgiveness. I dont expect her to accept my apology. I don't even expect her to even speak to me again. I know I'm not entitled to that at all. I dont deserve it tbh. That I just want her to be happy in life. That i wish the best for her. That I will always be here for her. That if she ever changed her mind about me she knows how to find me.

Is this a good idea? Me and this girl were deeply in love we were planning marriage and a life together. I do miss her a lot and want her back in my life. But I'm also aware that she may never come back and I accept that. I just want to right my wrongs as best as I can.
Is it a good idea to send a hand written apology letter to an ex girlfreind that you hurt?
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