Cyclic Groups - Existence of the identity 
How is it claimed that the identity of a cyclic group $G$ with generator 'a' can be written in the form of $e = a^m$ where $m$ is a positive integer. 

A proof for a subgroup of $G$ being in itself cyclic claims that any element in the subgroup can be written in the the form $a ^ n$ and assume that $n > 0.$ Furthermore the Well Ordering principle is used to claim the existence of $m$ above. 
The definition of a cyclic group only requires its members be able to be raised to any power of the generator. 
Suppose g is an element of $G$. Then $g = a ^ i$ where $i < 0$. How can I proceed to prove that g can be written in the form of a positive power of $a$?

First I assume that we're talking about a finite cyclic group.  The group $(\mathbb Z, +)$ is an example of what some people would call an infinite cyclic group and it is certainly not true that $n\cdot 1 = 0$ for some $n > 0$ (note everything is translated into additive notation in that example, but I switch back to multiplicative notation below).
Now assume the group is finite cyclic and generated by $a$.  Look at the sequence $(a, a^2, a^3, a^4, \ldots)$.  As the group is finite this sequence must repeat at some point, say $a^n = a^m$ for some $n > m$.  Now multiply both sides by $a^{-m}$ to get $a^{n - m} = 1$ and $n - m > 0$.
