Can a sequence contain an undefined term (e.g. a member acquired by dividing a number by zero)? I learned about the theorem that every convergent series is bounded.
However, take $a_n = \frac{1}{n-3}$ with $n=3$. 
It is unbounded, since $\sup{a_n} = +\infty$ (EDIT: it is not.). 
And yet, it is convergent with $\lim {a_n} = 0$
Am I violating some axiom of sequences with this one?

Your example is not a valid sequence, as it contains an undefined term.
