to send mail to someone repeatedly or to cover something thoroughly with mail.
From the prefix 'be-' (meaning to make or cover) combined with 'mail'. The 'be-' prefix became common in Middle English for creating verbs that meant to cover or affect something thoroughly.
The prefix 'be-' is a linguistic time machine—it was wildly popular in Middle English but fell out of fashion! Words like 'befriend' and 'besmirch' are relics of when English speakers loved adding 'be-' to nouns and adjectives to make action verbs.
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