How can we transition from |An+1 - An - L|<e to L-e<An+1 - An<L+e?

[transgalactic]

how they get from this expression

|An+1 - An - L|<e

to

L-e<An+1 - An<L+e

cant understand this transition
??

 

[symbolipoint]

how they get from this expression

|An+1 - An - L|<e

to

L-e<An+1 - An<L+e

cant understand this transition
??

Try to initially simplify your expression so it is easier to handle. Let v = An+1-An. Somehow, you probably mean something like A_n+1-A_n.
So now, you may write:

|v - L|<e

You can show that on a number line, knowing also that e>0, because it is greater than an absolute value, a positive number. You can also represent on the number line, -e, which is on the opposite side of zero, equadistant from zero as e.

You have two situations possible, but ultimately you may see that they form conjointedness.
Either v-L is positive, or v-L is negative.

Can you now work with the original statement more successfully?

 

[HallsofIvy]

|x|< a means -a< x< a. In particular, |v-L|< e means -e< v-L< e and, adding L to each part, L-e< v< L+ e.