I've never seen "I remains" at the bottom of a letter; I can't explain it.
However, an old-fashioned (and now mostly obsolete) way of ending a letter is e.g.
1. I remain your loving son
2. I remain your obedient servant
3. I remain your affectionate brother
"I remain" here means "I am still", and assures the recipient that the relationship is intact. (I would not recommend this ending in business or personal correspondence, though!)
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