how they get from this expression
|An+1 - An - L|<e
to
L-e<An+1 - An<L+e
cant understand this transition
??
We know that if $\displaystyle \left|\text{whatever}\right|<\varepsilon$ then it is also true (by definition) that $\displaystyle -\varepsilon<\text{whatever}<\varepsilon$. So applying this concept to your inequality we get $\displaystyle \left|A_{n+1}-A_n-L\right|<\varepsilon\implies -\varepsilon<A_{n+1}-A_n-L<\varepsilon$. Adding $\displaystyle L$ to the inequality gives $\displaystyle L-\varepsilon<A_{n+1}-A_n<L+\varepsilon$