I'd also like to add one more clarification: Cauchy sequences require _any two_ elements after N (the "certain point" mentioned above) to be within epsilon of e

I'd also like to add one more clarification: Cauchy sequences require _any two_ elements after N (the "certain point" mentioned above) to be within epsilon of each other. Why? I won't explain it fully, but hopefully a counterexample can convince you so: Take the common harmonic series (a_n = 1+1/2+1/3+...+1/n) Each successive element differs from the previous by 1/n. Therefore, informally we can see that their pairwise differences gets arbitrarily small. However, this sequence is _unbounded_: The infinite sum of the harmonic series tends to infinity. It therefore cannot be Cauchy. (Cauchy "bounces around a fixed point")