I am using Rudin's book on Functional Analysis. I am studying the proof that the space $C(\Omega)$ of continuous functions on an open set $\Omega \subseteq \mathbb{C}$ is a Frechet space. I encountered a detail that I can't understand and I hope you guys here can help me.
Here is a screenshot of the part:
Here, the $p_n$'s are seminorms defined by supremum of $f$ on a compact set $K_n$. I understand that the sequence $\{ f_i \}$ converges uniformly on each $K_n$ and its limit is continuous on $K_n$.
My problem is: how do we know that the limit $f$ is continuous on $\Omega$? In fact, I'm not sure why we get the same limit when we consider different $K_n$'s?
If we restrict $f_{i}$ to the interior of $K_n$, then it converges uniformly there to $f\mid_{K_n}$, so $f\mid_{K_n}$ is continuous on the interior of $K_n$, and since continuity is a local property, $f$ is continuous on all of $\Omega$.
To see that you get the same limit when considering different $K_n$, so that we have a well-defined limit on all of $\Omega$, suppose that $f$ and $f'$ are the limits of $f_i$ considered on the compact sets $K$ and $K'$. We wish to show that these are compatible, so they agree on $K \cap K'$. But this is clear, for suppose that they did not agree at $x \in K \cap K'$, i.e. $f(x) \neq f'(x)$. Then the sequence $f_i$ cannot both converge to $f$ on $K$ and $f'$ on $K'$ in the appropriate metrics, because they cannot both converge pointwise at $x$.