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I ask this question only to those who read or have this book:

If you have Baby Rudin, it would be even better.

On the page 34 of the text Conway's Functions of One Complex Variable Vol 1, it proves the Chain Rule

but it seems the proof is not valid:

It uses sequences to show the limit is satisfied to be the differentiation i. e., f'(g(z_0))g'(z_0)

We all know for the limit of function to hold, sequence should be chosen arbitrary

For example, to show limf(x) = f(p) with x to p, choose arbitary x_n such that x_n to p and then show limf(x_n)=f(p).

But in some parts of the proof

it uses constructed sequences which, obviously, not arbitrary.

For example, in Case 1 of the proof, when deriving f'(g(z_0)), it uses a sequence f(x+h_n) which is not arbitrary, obvious, though h_n is arbitrary.

And in Case 2, it separate h_n to l_n and k_n which are also not arbitrary.

So I think the proof is not really valid.

Am I the only one who think like this?

Btw can I use just Rudin's Chain Rule proof though this proof is on real-domain functions?

If you have Baby Rudin, it would be even better.

On the page 34 of the text Conway's Functions of One Complex Variable Vol 1, it proves the Chain Rule

but it seems the proof is not valid:

It uses sequences to show the limit is satisfied to be the differentiation i. e., f'(g(z_0))g'(z_0)

We all know for the limit of function to hold, sequence should be chosen arbitrary

For example, to show limf(x) = f(p) with x to p, choose arbitary x_n such that x_n to p and then show limf(x_n)=f(p).

But in some parts of the proof

it uses constructed sequences which, obviously, not arbitrary.

For example, in Case 1 of the proof, when deriving f'(g(z_0)), it uses a sequence f(x+h_n) which is not arbitrary, obvious, though h_n is arbitrary.

And in Case 2, it separate h_n to l_n and k_n which are also not arbitrary.

So I think the proof is not really valid.

Am I the only one who think like this?

Btw can I use just Rudin's Chain Rule proof though this proof is on real-domain functions?

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