- The problem statement, all variables and given/known data
Hi,
As part of the proof that :
the set of periods $\Omega_f $ of periods of a meromorphic $f: U \to \hat{C} $, $U$ an open set and $\hat{C}=C \cup \infty $, $C$ the complex plane, form a discrete set of $C$ when $f$ is a non-constant
a step taken in the proof (by contradiction) is :
there exists an $w_{0} \in \Omega_{f} $ s.t for any open set $U$ containing $w_{0}$, there is an $w \in \Omega_{f} / {w_0} $ contained in $U$
Now the next step is the bit I am stuck on
By the standard trick in analysis, we can produce a sequence of periods $\{w_n\}$ such that $w_{n} \neq w_{0} $ and $\lim_{n\to \infty} w_{n} = w_0 $
Relevant equations
The attempt at a solution
It's been a few years since I've done analysis, and the 'trick' has no name so I am struggling to look it up and find it in google.
A proof of this to understand it's meaning is really what I'm after , what's the idea behind the construction / significance in the usual context it would arise
I am also confused with the notation, does $n=0$ ? So the sequence converges to it's first term, or is $n$ starting from one
Many thanks in advance
The standard trick is: By assumption, there exists a period $\ne w_0$ in the open ball $B_{1/n}(w_0)$. Let $w_n$ be any such period. Then clearly $w_n\ne w_0$ for all $n$; and $|w_n-w_0|<\frac 1n$ implies $w_n\to w_0$.
Is suppose, one could do the whole proof simpler: As $f$ is assumed non-constant, we can pick $z_0$ such that $f$ is not constant in any neighbourhood of $z_0$. In particular, there exists $r>0$ such that $f(z)\ne f(z_0)$ for all $z$ with $0<|z-z_0|<r$. For otherwise we could find a sequence $z_n\to z_0$ (with all $z-n\ne z_0$) with $f(z_n)=f(z_0)$; by the Identity Theorem, this would imply that then $f$ is constant in a neighbourhood of $z_0$, contradiction. But if $f(z)\ne f(z_0)$ for all $z$ with $0<|z-z_0|<r$, then there cannot be a period $w$ with $0<|w|<r$. As the set of periods is an additive subgroup of $\Bbb C$, it must be discrete (namely, all elements are isolated points with distance at least $r$ to any other period).