Is it appropriate to use demise in the following sentence:
it is with deep sympathy that we announce the sad demise of ...
It sounds archaic and I was not sure whether it was used correctly. How is demise usually used?
It is entirely appropriate to use demise in that sentence. Statements of sympathy and condolence are normally of a rather solemn and formal nature, and the higher register is entirely suitable to the occasion. Demise is also a less direct term than death: the use of euphemism, however minor, softens the bald statement of what may be to some a tragic passing.
The use of demise is correct in the given context. Just as one might say 'passing away'.
It is not archaic but formal and somewhat euphemistic. In an obituary announcement, demise is used rather than death — so as not to hurt sentiments.
demise noun [in singular] (OxfordDictionaries)
1 a person’s death: Mr Grisenthwaite’s tragic demise
the end or failure of an enterprise or institution: the demise of industry
Wikipedia notes that it is a euphemism
Through euphemism, "(a person's) demise" is often used as a stilted term for a person's death.
1911 Encyclopædia Britannica records:
"Demise" is thus often used loosely for death or decease.
Note the word often in both the later references.
We can’t imagine the grief you must be suffering from his demise.
I am completely shocked at the sudden demise of your brother. Please accept my sincere condolence.
Coping with the emotional turmoil after the demise of a family member or friend is a heart wrenching experience.
Demise can be used to describe the death of a person (perhaps especially in the case of a famous yet not-necessarily-beloved person, or in a tragic and untimely death), or to the crumbling downfall of an institution. I believe the latter usage is more common in modern writing, but I couldn't say that for sure (although the usages listed on the Wordnik page would seem to back me up).
That said, I don't know if I'd go so far as to call your example usage archaic. It's certainly within the scope of the dictionary's definition, and it's not hard to find such usages in a Google book search:
The deathblow had occurred as he rolled up the front of the truck and bounced, then met his demise on the upturned tailgate of the pickup truck. (from Angel of Death – Downfall, by H. Edward Newton, 2004)