Spivak's proof of the chain rule in $\mathbb{R^n}$. The proof can be found on page 19.
I'm confused by the last step. "Equation 6 now follows easily."
Spivak's proof of the chain rule in $\mathbb{R^n}$. The proof can be found on page 19.
I'm confused by the last step. "Equation 6 now follows easily."
I thought that I took this proof from Spivak. You might find it more clear.
Divide across by $|x-a|$ giving $\frac{|\psi(f(x))|}{|x-a|} \leq \epsilon \frac{|\phi(x)|}{|x-a|}+\epsilon M$. Line (4) shows that the $\frac{|\phi(x)|}{|x-a|} \to 0$, hence bounded, so this shows that $\frac{|\psi(f(x))|}{|x-a|} \to 0$, since $\epsilon >0$ was arbitrary.