I would like to understand and answer the following question from Serge Lange's Introduction to Complex Analysis at a graduate level.
I understand how is one supposed to use the hint to prove the desired result but I do not understand how one proves the hint. (I think once the hint is proven we have basically shown that about every point the function is locally constant, even about the poles, so the poles are removable singularities and hence the function is constant).
I tried setting up equations of the lines $$aw_1+bw_2+cw_3=\epsilon$$ in complex and vector forms but it was a dead end as I wasn't able to prove that such $a,b,c$ exist for all epsilon not equal to zero.
Does anybody have any ideas has what the construction of such proof would look like and moreover, how should one think about this problem intuitively?
Case 1: $w_1/w_2 \not\in \Bbb R.$ Consider $\Bbb C$ as a 2-dimensional vector space over the field $\Bbb R.$ So $\{w_1,w_2\}$ is a vector-space basis for $\Bbb C.$ So there exist $r_1,r_2\in\Bbb R$ such that $r_1w_1+r_2w_2=w_3.$
Definition. For $x\in \Bbb R$ let $d(x,\Bbb Z)=\min \{|x-z|:z\in\Bbb Z\}.$
Lemma. For any finite $S\subset \Bbb R$ and any $\epsilon >0$ there exist infinitely many $n\in\Bbb N$ such that $\forall x\in S\,(d(nx,\Bbb Z)<\epsilon).$
We only need this lemma when $S$ has 1 or 2 members, and we do not need infinitely many $n\in \Bbb N$. Just one $n$ will be needed for a given $\epsilon$.
Return the the 1st paragraph. By the Lemma, with $S=\{w_1,w_2\}$, for any $\epsilon >0$ there exists $n_{\epsilon}\in\Bbb N$ such that $n_{\epsilon}r_j=m_j+\delta_j$ for $j\in \{1,2\}$, with $m_j\in\Bbb Z$ and $|\delta_j|<\epsilon.$ Hence $$|m_1w_1+m_2w_2-n_{\epsilon}w_3|=|n_{\epsilon}(r_1w_1+r_2w_2-w_3)-(w_1\delta_1+w_2\delta_2)|=$$ $$=|0-(w_1\delta_1+w_2\delta_2)|\le$$ $$\le (|w_1|+|w_2|)\epsilon$$ which can be arbitrarily small but not $0$, because if $m_1w_1+m_2w_2-n_{\epsilon}w_3=0$ then $w_1,w_2,w_3$ would be linearly dependent over $\Bbb Q.$
Case 2. $w_1/w_2\in \Bbb R.$ By the Lemma with $S=\{w_1/w_2\},$ for any $\epsilon>0$ there exists $n_{\epsilon}\in \Bbb N$ such that $n_{\epsilon}(w_1/w_2)=m+\delta$ with $m\in\Bbb Z$ and $|\delta|<\epsilon.$ Hence $$|n_{\epsilon}w_1-mw_2|=|\delta|\cdot |w_2|\le\epsilon |w_2|$$ which can be arbitrarily small but not $0$, because if $n_{\epsilon}w_1-mw_2=0$ then $w_1,w_2$ would be linearly dependent over $\Bbb Q.$