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"Ultimate", "penultimate", "proximate", and "approximate" are English words.

"Ultimate" is common, partly because it is used in advertising.

"Penultimate" is less common than "ultimate", partly because nobody wants to advertise that their product is second-best.

"Approximate" is common, both as an adjective and as a verb. The adjective "approximate" is more formal than the adjective "rough"; the verb "approximate" is more formal than "make a careful guess" or "come close to".

"Proximate" is not very common. Both "penultimate" and "proximate" are sometimes used by educated writers.

Ultima and proxima are so rare that they might as well still be Latin words. Until I looked them up just now, I had no idea that ultima specifically referred to the last syllable in a word, instead of generally the "furthest out" thing in a group of things. The most common use of ultima is in the title of the on-line game Ultima Online. The most common use of proxima is in the name of the second-closest star to the Earth, Proxima Centauri.

Those both exist as words in English, but with very specialised meanings or very infrequent use. These relate to their meanings in Latin, which are reflected in modern Spanish (and possibly Portuguese, too), albeit with accents on some letters.

Ultima is used in specialised situations to refer to the last syllable of a word.

Proximo, the nearest any attested English word comes to proxima without being a proper noun, is a little-known term that means "of (the) next month", for example "on the 3rd proximo" means "on the third day of next month". Ultimo is correspondingly used (or not) for the previous month.

These are both rather obscure words and most people are not familiar with them. I only know what they mean now because I thought I should look them up before saying "those are not English words".

Don't use them. Well, use ultima if you're in the sort of context where other people use it. Don't use it otherwise, and basically no-one uses proxima unless they're trying to show off their expansive vocabulary.

I have noticed that the words ultima and proxima is rarely used.

How native speakers use these?

I know from my knowledge of classical Latin and of Spanish that these words mean 'last' and 'next' respectively.

How do native speakers use these? They don't. I have never used or heard or read these words in English.

Edit 1

I see from @SamBC that these words appear in English dictionaries with very specialised meanings. I didn't know those meanings and I will not make any attempt to remember them.

Edit 2

It occurs to me that astronomers use 'proxima' as in Proxima Centauri, a red dwarf star, in the constellation of Centaurus. This is however the name of the star and is not used in general conversation by most people unless they happening to be discussing that particular celestial body.

In 18th and 19th century business correspondence, these words, invariably abbreviated as "Ult." and "Prox." were often used to mean mean 'of the last month" and "of the next month". Such a letter might run:

In your favor of the 19th ult., you proposed delivery on the 20th prox. The supplies are needed by the 10th prox. at the latest.

(Here "favor" was a polite term for "letter addressed to me" now also obsolete.)

I have not seen this usage in any document later than about 1920, and rarely in any later than 1880. Currently I see them only in historical fiction of that period, or in the star name Proxima Centauri mentioned in other answers.

Both terms are very solidly examples of jargon (in any sense of the term actually, depending on the context).

Ultima

'ultima' could be:

• A linguistics term, with the exact single meaning listed in both the OED and Webster's.


• The name of a video game series.


• The name of a spell in the Final Fantasy video games.


• Part of the name of any of a couple of astronomical objects (Ultima Thule being the trivial example).


• Possibly used as part of other terminology in a different field of science, engineering, or mathematics.

There are a handful of other words in English with varying meanings that are derived from the same Latin root as 'ultima':

• 'Ultimate': Pretty widely used in advertising and branding. Refers to the absolute best version, form, or iteration of something.


• 'Penultimate': Not very widely used outside of academic settings. Refers to the second-to last instance of something.


• 'Ultimatum': Contextually common. Used to refer to a truly final offer in negotiations (or similar situations), typically one which if rejected will immediately end negotiations.

There are probably others I've missed.

Proxima

I've actually not found any definition searching online for this other than the use in astronomy as a clip form of the name of the star Proxima Centauri (MWD doesn't list it, I don't have an OED subscription so I can't check there, dictionary.com lists the astronomy use, and everywhere else I've checked matches either MWD or dictionary.com). I've not seen it used for any other purpose either (unlike 'ultima', which I've seen multiple times before).

There are, however, also a number of other English words derived from the same Latin root:

• 'Proximal': Used mostly in scientific settings, especially biology and medicine. Usually used as an antonym to 'distal', occasionally as a synonym for 'proximate'.


• 'Proximate': Also mostly in scientific settings, referring to an object or event that directly follows or precedes a specific reference point in a sequence.


• 'Proximo': Rarely used except in prose. This is a direct clip of the Latin phrase 'proximo mense', literally 'in the next month', and retains the same meaning.


• 'Approximate': Widely used in mathematics and science, as well as some usage in general speech in it's various forms. All forms ultimately are variants on the noun 'approximation', and refer to something being close to some other reference point (usually a correct answer to a problem).

I've probably also missed some here too.

Ultima Thule and Proxima Centauri are English in the same way that Jupiter or Homo Sapiens are English.

They are familiar scientific names for specific things, and would be recognised as such by an English speaker, but they are not English words in and of themselves.

Most astromomical names are derived from Greek, Latin or other ancient languages and are explicitly not English because they are intended for international use.

In the case of the two words you asked about, "Ultima" and "Promima" do apparently exist in English language dictionaries as words, but are not in common useage as stand-alone words and where they are used, the meanings of those words in English are not the same as the Latin meanings intended by the astronomical names.

As a native English speaker, I have never heard those words use "in the wild", except under the following contexts:

Ultima:

• The astronomical body Ultima Thule
  


• Richard Gariot's computer game series Ultima.

Proxima:

• The star Proxima Centauri.

While you might find some definitions for them in some dictionaries that say otherwise, the words don't really have any general meaning on their own for the common English speaker.