I am reading "Calculus" by Michael Spivak.
Spivak proved that the following inequality holds for sufficiently large $N$:
$$|\sum_{n=0}^\infty a_n \frac{((z+h)^n-z^n)}{h}-\sum_{n=0}^N a_n \frac{((z+h)^n-z^n)}{h}| < \frac{\epsilon}{3}.$$
But I think this inequality is obvious because
$$\lim_{N\to\infty} \sum_{n=0}^N a_n \frac{(z+h)^n}{h} = \sum_{n=0}^\infty a_n \frac{(z+h)^n}{h},$$
$$\lim_{N\to\infty} \sum_{n=0}^N a_n \frac{z^n}{h} = \sum_{n=0}^\infty a_n \frac{z^n}{h}.$$
I cannot understand what Spivak is doing.
Is this inequality really not obvious?
