Hi guys
Is that correct to use the indefinite article in this sentence?
“His daughter is married to _A_ man who lives by his pen” . Because “who lives by his pen” has more descriptive meaning then limiting, right?

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Yes. That is correct. The definite article would be correct in a few different cases:

If several men had been mentioned before this sentence. One man lives by building houses. One man lives by his sword. One man lives by his pen. One man lives by teaching. Then, the sentence would be:

“His daughter is married to the man who lives by his pen,” since that defines which man of the various ones describes.

Another way of thinking about it is: since several men are authors (who live by their pens), using the definite article in the sentence would imply only one author exists in the world… which is not the case.