I don't get what you mean by method extends another method. But I get the point that we are talking about HttpServlet here so overring the service method is not required and should not be done. The service method for HttpServlet knows which one to call between the doGet and doPost. To make them do the same thing then your method of making a separate method and calling it from both is probably best.
Yea, you are right, I got lost, what I said about doGet and doPost is applicable for service method. I mean service method overrides generic service method, and do all the preliminary things about an HTTP request and calls the appropriate doXXX method. So we should not override service method, else we would have to do all the preliminary work ourselves.