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I have to use only the following defn to prove the limit exists.

Using def (2) sec 15. prove the following.

Section 15 (2) =$ |f(z) - w_0 |< \epsilon \space \space \space $ Whenever $ 0< | z-z_0 | < \delta $

1) $ \lim_{z \to z_{0}} Re(z)= Re(z_0)$

$ \lim_{z \to z_{0}} |f(z) - w_0 |= z= (x+iy)$ Re(z) = x $ \space \space \space \lim_{x \to x_{0}} |x - x_0 |< \epsilon $

i define $|x - x_0 |<1$ and $|x| - |x_0 | \leq |x - x_0 |$ thus $|x| < 1 + |x_0 | $

This implies that $ |1 + |x_0 | - x_0 |< \epsilon $

but $-x_0 \leq |x_0|$ so $ |1 + 2|x_0 | | \leq |1| + |2|x_0 | |= 1 + 2|x_0 | < \epsilon $

let $\delta = \min \{ 1,\frac { \epsilon }{1 + 2|x_0 |} \} $

does this work? There are no worked out examples in my textbook and i have never taken analysis.

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    "Section 15 (2) =|f(z)−w0|<ϵ |f(z)−w0|<ϵ Whenever 0<|z−z0|<δ" This statement cannot be parsed by my brain. When writing math it's better to use complete sentences with correct grammar and punctuation. At the risk of sounding snarky, which is not intended, are you claiming that Section 15 (2) is equal to $| f(z) - w_0 |$? But Section 15 (2) is not a number, it's a section in a book, so it can't be equal to a number $| f(z) - w_0|$. – 2017-01-29
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    it doesn't actually say the definition in the question i had to look it up and thats what it says in my textbook word for word. man i hate analysis, i really need a new textbook :( – 2017-01-29
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    No worries. I'd guess that definition (2) in section 15 was supposed to say: "The statement that $\lim_{z \to z_0} f(z) = w_0$ means that if $\epsilon > 0$ then there exists $\delta > 0$ such that if $0 < |z - z_0| < \delta$ then $| f(z) - w_0| < \epsilon$." Or something equivalent to that. – 2017-01-29

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