I'm reading the chain rule proof from the vector calculus supplement to Marsden and Tromba. As part of the proof, to a supplementary theorem, they say, choose $\epsilon = 1$. I have attached the image.
I am confused here because I thought in $\epsilon-\delta$ definitions, you could only choose $\delta > 0$, and that your proof must work for all $\epsilon$.
Thank you.
