The union of a sequence of countable sets is countable.

Question

While working on the theorem below, I constructed the following proof:

Theorem. If $\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ is a sequence of countable sets, then

$$ \bigcup_{n\in\mathbb N}E_{n} $$

is countable.

Proof. Let $S=\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ be a sequence of countable sets. Moreover, let $x_{n,m}$ be the $m$th element of the $n$th set in $S$. Construct a sequence of sequences

$$ \mathcal S=\left\langle\left\langle x_{n-m+1,m}\right\rangle_{m\in\mathcal N_n}\right\rangle_{n\in\mathbb N}, $$

where $\mathcal N_{n}=\left\{k\in\mathbb N:k\leqslant n\right\}$. Then $\mathcal S$ contains all the elements of

$$ T=\bigcup_{n\in\mathbb N}E_{n}. $$

Observe that $\mathcal S$ is a surjection $\mathbb N\to T$, because for every $n\in\mathbb N$, $\mathcal S$ yields a finite, and thus countable, subsequence. Furthermore, because $\mathcal S$ may contain duplicate elements, an $U\subseteq\mathbb N$ can be found such that $\mathcal S$ is an injection $U\to T$. We have then found a bijection $\mathcal S:U\to T$. Hence, $T$ is countable. $\blacksquare$

Does it sound convincing?

Edit: I just realized that I cannot pinpoint duplicates because of how $\mathcal S$ is constructed, can I? :-(

Answer 1

You’re trying to list $T$ as $$\langle x_{1,1},x_{2,1},x_{1,2},x_{3,1},x_{2,2},x_{1,3},\ldots\rangle\;,\tag{1}$$ but instead you’ve constructed the sequence

$$\Big\langle\langle x_{1,1}\rangle,\langle x_{2,1},x_{1,2}\rangle,\langle x_{3,1},x_{2,2},x_{1,3}\rangle,\ldots\Big\rangle\;,$$

a related but definitely different animal. It’s actually easier to say what $n\in\Bbb N$ corresponds to $x_{k,\ell}$ in the list $(1)$ than to go in the forward direction. Count the elements of $(1)$ that precede $x_{k,\ell}$. They certainly include all $x_{i,j}$ such that $i+j<k+\ell$, and there are

$$\sum_{n=2}^{k+\ell-1}n=\frac{(k+\ell-1)(k+\ell)}2-1$$

of those. (The summation starts at $2$ because we always have $i+j\ge 2$.)

Let $m=k+\ell$; it also includes all $x_{m-i,i}$ such that $1\le i\le\ell$, and there are $\ell-1$ of those. Thus,

$$\frac{(k+\ell-1)(k+\ell)}2-1+\ell-1=\frac{(k+\ell-1)(k+\ell)}2+\ell-2$$

terms of $(1)$ precede $x_{k,\ell}$, and $x_{k,\ell}$ is therefore term number

$$\frac{(k+\ell-1)(k+\ell)}2+\ell-1$$

of the sequence $(1)$.

Answer 2

Proposition 1: A set $X$ is countable if and only if there exists a surjection $\tau: \mathbb N \to X$.
Proof: See link.

For each countable set in $\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$, select a surjection $\tau_n: \mathbb N \to E_{n}$ and let $X$ be the union of the $E_n$.

Proposition 2: There exists a surjection $\gamma: \mathbb N \times \mathbb N \to X$.
Proof
Define $\gamma$ by $(n,m) \mapsto \tau_n(m)$.

Cantor's diagonal argument gives us

Proposition 3: There exists a bijection from $\mathbb N$ to $\mathbb N \times \mathbb N$.

Since the composition of two surjective maps is also surjective, we have shown that the union of the $E_n$ is a countable set.

Answer 3

Perhaps one can be more concrete. Assume that the indexing of your $x_{n,m}$ starts at $1$. Let $p_n$ be the $n$-th prime. Map $p_n^m$ to $x_{n,m}$, and the numbers not of the form $p_n^m$ to any fixed object. This gives a surjection from the natural numbers to our union.