On a math exam recently, the students were asked to use the definition of a limit of a sequence to prove that the sequence given by 3n/(3n+5) converges to 1. Given a positive number Ɛ, the definition requires proving the existence of some number N such that if n>N then |3n/(3n+5) - 1|<Ɛ.
As a consequence of the definition, once a sufficiently large N is found, any larger value of N will also suffice. Many students set |3n/(3n+5) - 1|=5/(3n+5)<Ɛ and solved for n to find N = (5-5Ɛ)/(3Ɛ). However, the professor decided to include an extra step: 5/(3n+5) < 5/n <Ɛ, which leads to another sufficient value N = 5/Ɛ.
Although most students gave a correct proof (consistent with the definition in their book), the lecturer took off points because they didn't find the "best" value of N. The lecturer claims that the author would have used some (unnecessary) inequalities to find the "better" N, which is probably true.
When students complain about losing points, I tell them that their answer is correct and that they should seek full credit for their work. The lecturer suggests that I am putting the students in a position in which they may "pick a side" and that ultimately the lecturer is in charge.
Who's wrong here?
Update: I was not notified about the lecturer's decision to remove points until after I gave the midterms back to the class. Once students started asking me about the missing points, the only written justification left by the lecturer was "not best N."
By "best N," the lecturer was referring to the N value found by using the additional inequality 5/(3n+5) < 5/n <Ɛ. By "best," he does not mean "smallest" (and by definition, there is no largest N).