A university education

There is a sentence that reads : William Golding was a very famous English write. After a university education, the first job he did was to work in the theatre as an actor. I know that this sentence is grammatically correct, but a friend of mine is not convinced of the underlined part of the sentence. He believes that this part is wrong as the use of the indifinte article here is not proper . I tried to explain to him that the use of "a" is ok since there are various kinds of education and the university education is one of them. It seems that I didn't get my idea across to him. Do you have a convincing answer to my friend's question?