Proof chain rule; question about specific step

Question

How do they come up with (5)? Why don’t they just write $$ \left\vert\frac{(g\circ f)(x_n)-(g\circ f)(a)}{x_n-a}\right\vert\leq C? $$ I don't understand where the factor $\left\vert\frac{f(x_n)-f(a)}{x_n-a}\right\vert$ comes from. I can't be from this: $$ \left\vert\frac{(g\circ f)(x_n)-(g\circ f)(a)}{f(x_n)-f(a)}\right\vert\cdot\left\vert\frac{f(x_n)-f(a)}{x_n-a} \right\vert, $$ because for all we know $f(x_n)=f(a)$. So could someone explain this to me?

Answer 1

You have to consider two cases. If $f(x_n)=f(a)$ then $g\circ f(x_n)=g\circ f(a)$ and so (5) holds since both sides are zero. If $f(x_n)\ne f(a)$ then you can divide by $f(x_n)-f(a)$ and use the inequality above (5).