factor group is cyclic. Prove that a factor group of a cyclic group is cyclic.

I didn't understand last two lines of proof ..
Therefore $gH=(aH)^i$ for any coset $gH$.
so $G/H$ is cyclic , by definition of cyclic groups.
How $gH=(aH)^i$ of any coset $gH$.
proves factor group to be cyclic. 
Please explain.

I just wanted to mention that more generally, if $G$ is generated by $n$ elements, then every factor group of $G$ is generated by at most $n$ elements:
Let $G$ be generated by $\{x_1,\ldots x_n\}$, and let $N$ be a normal subgroup of $G$.  Then every coset of $N$ in $G$ can be expressed as a product of the cosets $Nx_1,\ldots, Nx_n$.  So the set $\{Nx_1,\ldots,Nx_n\}$ generates $G/N$, and this set contains at most $n$ elements. 
(Note that the cosets $Nx_i$ will not all be distinct if $N$ is non-trivial, but it's fine to write the set this way, just as $\{x^2 \mid x\in \mathbb{R}\}$ is a perfectly valid description of the set of non-negative real numbers.)
The result about cyclic groups is then just the special case $n=1$ of this.
