My Real Analysis course uses the $\epsilon - n_0$ definition of the limit, but I have noticed that the $\epsilon - \delta$ approach seems to be more common. Could someone please explain both formally and informally the difference between the definitions? Is there an advantage to using one over the other? An example of a simple proof using both definitions would be great!
(Sorry if this is a silly question. Looking at the definitions, I don't think that $n_0 = \delta$. but I could be wrong. If $n_0 = \delta$ then at least there is a simple answer, though I'll feel pretty embarrassed.)