After reading Rudin's proof, using a diagonal argument, that a union of countable sets is countable, I'm trying to understand why it wouldn't be possible to adapt the argument to an uncountable collection of countable sets, which isn't in general countable. I have a conjecture as to why that's the case, but I'll sketch his argument first.
Let $\{E_n\}$ for $n=1,2,3, \ldots$ be a collection of countable sets. As each $E_k$ is countable, we can list its elements as the sequence -- in particular, the image of a bijection $\mathbb{N} \to E_K$. So $E_k = \{x_{k1}, x_{k2}, x_{k3}, \ldots\}$. We then arrange these sequences in an infinite array: \begin{align*} x_{11}, x_{12}, x_{13}, \ldots \\ x_{21}, x_{22}, x_{23}, \ldots \\ x_{31}, x_{32}, x_{33}, \ldots \\ \vdots \end{align*} Via a diagonal process, we assign a natural number to each $x_{ij}$, giving a bijection between $S$ and a subset of $\mathbb{N}$, so $S$ is at most countable. As $S \supset E_1$, where $E_1$ is infinite, $S$ must be countably infinite.
I believe the step that breaks down the second the collection becomes uncountable is the fact that I arranged the sequences in an array. I have "countably many rows," where each row is in effect indexed by a natural number. If I have uncountably many rows, I don't think I can list the sequences in an array or describe any well-defined diagonal process to capture all of them.
Is this the correct idea for why the argument fails for an uncountable collection?