In scholarly writing, this is standard form. I have always assumed that the present tense indicates that the proof is to be demonstrated in the present, which is to say: as we read it from this point forward. Linguists may well employ a specific term for this use of the present tense, such as immediate present.
As pointed out in another answer, English verbs do not inflect to express action in the future (as many other languages do) and English thus lacks a true future tense. Nevertheless, we must, can, and do talk about actions in the future, e.g. by combining the auxiliary verbs "will" and "shall" with the unmarked infinitive; and it is common (and useful) to refer to such usage colloquially as the future tense.
This ersatz future tense is occasionally seen in an introductory paragraph, of the form:
In what follows, we will prove...
In an abstract, the present is most commonly used.
Note that there are many manuals of style which address this and other aspects of writing within a given discipline.
A useful summary is presented in answer to the question "In what tense (present/past) should papers be written?" at our sister site.