Sorry, I incorrectly typed this into Latex. I meant to say that the sequence n_i is a strictly increasing sequence. It is the sequence of all n for which n*x_n is greater or equal to epsilon. Progressing along the natural numbers, n(1) must be less than n(2), and n(i) < n(i+1). The contradiction arises when the decreasing x_n sequence: x(i) > x(i+1), implies that n(i) > n(i+1).
That is, 1/n(i) > 1/n(i+1) implies n(i) < n(i+1), a contradiction.
My rationale for doing it this way instead of a modified harmonic series proof, is that don't know any values of the n_i sequence. I feel as though I need actual numbers to show that a series is unbounded. The harmonic series proof of divergence uses the fact that we can group terms in such a way that we have a constant unbounded sequence of 1/2.
I am guaranteed the existence of n's from the statement, "there exist n such that n*x_n is greater or equal to epsilon". But I gain no numerical information to show that the series of those n is unbounded. What do you think of this?
I will upload a corrected PDF soon