What is the proper way to complete this proof of this simple limit?

[stark galleon]

Could you please explain the proper format of proving this limit? I have attached what I have written so far.

[median flower]

You have the assumption $0 < \abs{x-1} < \delta.\$
Now for $\abs{x+1}$ you would want to bound it given the assumption, for example [ 0 < \abs{x-1} < \delta \Rightarrow \abs{x+1} < c ] where $c$ is some positive real number.$\$
So you would have to restrict $\delta$ for example pick $\abs{x-1} < \delta = 1.\$
Then you get
[ -1 < x-1 < 1 \Leftrightarrow 1 < x +1 < 3 \Rightarrow \abs{x+1} < 3 ]