fixed point in regular expressions
As all productions in  $A\rightarrow \mathrm{ab}A\text{ }|\text{ }A\mathrm{b}\text{ }|\text{ }ε$ are linear, and (crucially) there is only one non-terminal symbol, all derivation sequences can be reordered into equivalent ones where left-side derivations always precede (or always succeed) right-side ones.

For instance:

$wAv \Rightarrow  w\mathrm{ab}Av \Rightarrow  w\mathrm{ab}A\mathrm{b}v \Rightarrow  w\mathrm{abab}A\mathrm{b}v \Rightarrow  w\mathrm{abab}A\mathrm{bb}v\equiv \\wAv \Rightarrow  wA\mathrm{b}v \Rightarrow  wA\mathrm{bb}v \Rightarrow  w\mathrm{ab}A\mathrm{bb}v \Rightarrow  w\mathrm{abab}A\mathrm{bb}v$ 

(This is a hint of what an inductive proof using fixed-points would look like in this particular case. Check this if you are determined to go down that road).

This means that this grammar can be converted into a regular grammar in two steps, by grouping left-linear and right-linear productions: 

1.


$A \rightarrow A'A''\\A' \rightarrow \mathrm{ab}A'\text{ }|\text{ }ε$
$A'' \rightarrow A''\mathrm{b}\text{ }|\text{ }ε$  

2.


$A \rightarrow A'A''\\A' \rightarrow \mathrm{ab}A'\text{ }|\text{ }ε$
$A'' \rightarrow \mathrm{b}A''\text{ }|\text{ }ε\\$ 

So $\mathcal{L}(A) = (\mathrm{ab})^*\mathrm{b}^*$