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I have always struggled with Analysis... I understand the concepts generally I feel like, but when I go to show something or try and prove something I feel my symbolic notation is awful. I know that

$$\left|{{\frac{f(z)-f(z_0)}{z-z_0}-f'(z_0)}}\right|\le\epsilon$$

implies

$$|f(z)-f(z_0)-(z-z_0)f'(z_0)|<\epsilon|z-z_0|$$

But why is it okay to drop the absolute value signs below, which was presented to be true in a chapter I'm reading...

$$f(z)=f(z_0)+(z-z_0)f'(z_0)+\eta\cdot(z-z_0)$$

with $|\eta|=|\eta(z)|<\epsilon$.

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    Do you mean to finish with "$\leq \epsilon |z - z_0|$ on the "implies"? – 2017-02-26
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    oh yes.... sorry. I fixed it. – 2017-02-26
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    both sides are positive – 2017-02-26
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    define eta such that the equality holds for eta positive, then you can leave the abs values – 2017-02-26

1 Answers 1

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Rafael Wagner has the idea right, but it is cleaner to approach it in this fashion:

Define $\eta = \frac{f(z)-f(z_0)}{z-z_0}-f'(z_0)$. (Note that $\eta$ is a function of $z$ and $z_0$.)

Then $$f(z) = f(z_0) + f'(z_0)(z - z_0) + \eta(z - z_0)$$

And by the existance of $f'(z_0)$, we know that for any $\epsilon > 0$, when $z$ is sufficiently close to $z_0, |\eta| < \epsilon$.

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    that was great. thank you! – 2017-02-26