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Why is the pointwise limit of a sequence of open functions always open?
If we have a sequence of open functions $(f_n)$ such that $f_n$ converges to $f$ pointwise, i.e. $\forall x \in E$ $f_n(x) \rightarrow f(x)$, then $f$ is also open. Why is this true?
A:
$f_n$ is open $\iff$ $\forall x\in E$ $\forall \delta >0$ $\exists N\in \mathbb{N}$ such that $f_n^{ -1}[B(x,\delta ) ]\subseteq U$
We need to show $\implies$
We have $\forall x\in E$ $\forall \delta >0$ $\exists N\in \mathbb{N}$ such that $f_n(x)\in B(f(x),\delta )$ (proving $(f_n)$ is Cauchy) $\implies$
We have $\forall x\in E$ $\forall \delta >0$ $\exists N\in \mathbb{N}$ such that $f_n^{ -1}[B(f(x),\delta )]\subseteq U$
What is the pointwise limit of $f_n^{ -1}[B(f(x),\delta )]$?
What is the pointwise limit of $B(f(x),\delta )$?
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![Dependence of the laser-induced trap depth $E_\mathrm{d}^\mathrm{L}$ on the trap laser power for potassium and argon.[]{data-label="fig:potassiumvsArgon"}](potassiumvsArgon.pdf){width="\columnwidth"}
![Dependence of the laser-induced trap depth $E_\mathrm{d}^\mathrm{L}$ on the trap laser power for potassium (green) and argon (blue). The orange line indicates the calculated trap depth based on Eq. .[]{data-label="fig:diametervsPressure"}](diametervsPressure.pdf