prove oneself vs demonstrate

prove oneself and demonstrate both are verbs.

Word	Noun	Adjective	Verb	Adverb
prove oneself	No	No	Yes	No
demonstrate	No	No	Yes	No

As verbs, demonstrate is a hypernym of prove oneself; that is, demonstrate is a word with a broader meaning than prove oneself:

prove oneself: show one's ability or courage
demonstrate: establish the validity of something, as by an example, explanation or experiment

Other hypernyms of prove oneself include establish, prove, shew, show.

prove oneself (verb)	demonstrate (verb)
show one's ability or courage	establish the validity of something, as by an example, explanation or experiment
	provide evidence for; stand as proof of; show by one's behavior, attitude, or external attributes
	give an exhibition of to an interested audience
	march in protest; take part in a demonstration