Precise definition of a limit: Why is the delta chosen to be 1 and not \sqrt{2}-1

Question

We have the following limit to prove using delta-epsilon

$\lim_{x \to 1} \frac{x+1}{1+\sqrt{x}}$

It can be manipulated into

$|x-1| |\sqrt{x}-1||\frac{1}{{(1+\sqrt{x})}^{2}}|<\varepsilon$

if our next step is to replicate $\sqrt{x}-1$ in

$-1<|x-1|<1$

should it not be

$0<|\sqrt{x}-1|<\sqrt{2}-1$

instead of the given answer of $0<|\sqrt{x}-1|<1$ ?

Problem and given solution here.