Why doesn't the proof need a specific value for δ in the ε-δ definition of continuity?

[Andrax]

Homework Statement

let f be a function t: lf(x)l≤lxl
show that f is continious at 0

The Attempt at a Solution

it's easy to see that f(0)=0
now $\forall$E>0 $\exists$α>0 $\forall$x$\in$D: lxl<α => lf(x)-f(0)l<E now in the solution manual they just put it like this : since lxl<α implies lf(x)-(f(0)=0)l<E then f i s continious at a , what I'm not getting is that they didn't give alpha a value they just want from x<alpha to the result ? I've been studying limits since 2012 so this is a weird issue to me , please help

 

[mfb]

Just take α=E.

 

[Andrax]

thanks i forgot a bit about the first definition that's why i had trouble ,i've got it now