in correspondence, etc., "in or of the next or coming month," noting a day in the coming month (proximo mense), Latin ablative singular of proximus "nearest, next" (see proximate). Often abbreviated prox. Compare ultimo, instant (adj.).
1590s (implied in proximately), "closely neighboring; next, immediate, without intervention of a third," from Late Latin proximatus, past participle of proximare "to draw near, approach," from proximus "nearest, next; most direct; adjoining," figuratively "latest, most recent; next, following; most faithful," superlative of prope "near" (see propinquity). Meaning "coming next in a chain of causation" is by 1660s. Related: Proximately.
"in the month preceding the present," 1610s, common in abbreviated form ult. in 18c.-19c. correspondence and newspapers, from Latin ultimo (mense) "of last (month)," ablative singular masc. of ultimus "last" (see ultimate). Earlier it was used in the sense of "on the last day of the month specified" (1580s). Contrasted with proximo "in the next (month)," from Latin proximo (mense).
mid-15c., "now, present, of the moment, current," from Old French instant "near, imminent, immediate, at hand; urgent, assiduous" (14c.) and directly from Medieval Latin instantem (nominative instans), in classical Latin "present, pressing, urgent," literally "standing near," present participle of instare "to urge, to stand near, be present (to urge one's case)," from in- "in" (from PIE root *en "in") + stare "to stand," from PIE root *sta- "to stand, make or be firm."
Sense of "immediate, done or occurring at once" is from 1590s. Of processed foods, by 1912; instant coffee is from 1915. Televised sports instant replay attested by 1965. Instant messaging attested by 1994.
The word was used 18c.-19c. in dating of correspondence, meaning "the current month," often abbreviated inst. Thus 16th inst. means "sixteenth of the current month" (compare proximo, ultimo).
<a href="https://www.etymonline.com/word/proximo">Etymology of proximo by etymonline</a>
Harper, D. (n.d.). Etymology of proximo. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/proximo