It is well known that uniform convergence implies pointwise convergence. I have checked on this page (see for example 1 and 2), that locally uniform convergence does not necessarily imply uniform convergence. So, I am wondering if locally uniform convergence implies pointwise convergence in general. Somehow it seems trivial, I still need to convince myself.
Here is what I thought: let $(f_n)_{n\geq 1}$ be a sequence of functions defined on a subset $A$ of $\mathbb{C}$. Assume there exists $f:A\to \mathbb{C}$ such that $f_n\to f$ locally uniformly on $A$. (*) Then $f_n\to f$ uniformly on every compact subset of $A$. Then $f_n\to f$ pointwise on every compact subset of $A$. Does it imply that $f_n\to f$ pointwise on $A$?
Edit: After having read the comments: [...] (*) Then $f_n\to f$ uniformly on every compact subset of $A$. Let $x \in A$ be given. Then $\{x\}$ is a compact subset of $A$. Then $f_n\to f$ pointwise on $\{x\}$. Since $x$ is arbitrary, $f_n\to f$ pointwise for all $x\in A$.
But this is still inconsistent. See my comment below.