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Writing to publish is pretty self-explanatory. You are writing with the intent of publishing and selling what you write. But writing for legacy means that you are literally writing the story of your life so your legacy lives on. The story is for family. Those great-great-great grandchildren who see a photo of you and ask, "Mommy, who is that lady?" THEY are they ones you write a legacy memoir for...they are the ones that will be handed that book. It can include photos, copies of letters, scans of postcards, anything you feel pertains to the story of YOU.
Many of us think of our life like a shout into the wind. The only one that realizes it happened is you. But you don't give your life, your experiences enough credit. Someone somewhere can benefit from the times you fell and struggled to get up, because they are on the ground right now, trying to find the will to stand.
Don't underestimate the power of your own ability to inspire. Start writing your memories and life lessons. There is no better gift you can give those that love you.