For example, in a proof that shows $f(x) = \sqrt x$ is uniformly continuous on the positive real line, the proof goes like:
Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$....
Or to show that every Lipschitz continuous function is uniformly continuous
Let $\epsilon > 0$ be given, and $\delta = \epsilon$....
Do these people have a magic ball that let them see what the $\delta$ value is going to work?
I often find myself struggling coming up with the $\delta$ value after doing a bunch of inequalities on $|f(x) - f(y)|< \delta$ to make it less than $\epsilon$. How do people know what $\delta$ is going to be in the first line of their proof?