Is it grammatically correct to say "Un prof enseigne les/des élèves"?

No.

Choose either les or des.

Yes, but is either of them correct?

Yes. If you have a particular group of students in mind, use "les." If you mean students in general, use "des."

"Un prof enseigne à ces élèves" is better

Yes, it is grammatically correct to say "Un prof enseigne les élèves" or "Un prof enseigne des élèves" in French. Both sentences are grammatically correct but have slightly different meanings.

- "Un prof enseigne les élèves": This sentence translates to "A teacher teaches the students." Here, the definite article "les" is used before "élèves" to specify that the teacher is teaching specific, known students.

- "Un prof enseigne des élèves": This sentence translates to "A teacher teaches students." Here, the indefinite article "des" is used before "élèves" to indicate that the teacher is teaching some undefined or unspecified students. It suggests that the teacher is teaching a group of students, but not necessarily a specific group.

So, which sentence to use depends on whether you want to convey a general idea of teaching students (using "des") or a more specific idea of teaching particular students (using "les"). Both are grammatically correct and can be used in different contexts.