What is the meaning of Cantor's method?

Question

What is the meaning of this theorem?

If we have a double sequence $\{x_{i,j} \}$ of real numbers where $i,j \in \mathbb{N}$ and if it happened that for any $i$, say for $i=i_0$ and for any subsequence of $j_k$ we always find a convergent subsequence within the sequence $x_{i_0,j}$ then, for this particular subsequence that we have already chosen $j_k$ there exists a convergent subsequence for all $i$?

Is this correct?

Also, where can I find the proof of such theorem. Can you refer me to a textbook or something?

Answer 1

If every "row" has "many" convergent subsequences (that is, every subsequence of the row has a convergent subsubsequence), then there exists a subsequence of the "colums" that converges row-wise.

Note that to have "many" convergent subsequences in the sense above is equivalent to being bounded.

Answer 2

By hypothesis there is an infinite $N_0\subseteq\Bbb N$ such that $\langle x_{0,n}:n\in N_0\rangle $converges. Suppose that for some $k\in\Bbb N$ we have an infinite $N_k\subseteq\Bbb N$ such that $\langle x_{k,n}:n\in N_k\rangle$ converges; then by hypothesis there is an infinite $N_{k+1}\subseteq N_k$ such that the subsequence $\langle x_{k+1,n}:n\in N_{k+1}\rangle$ of $\langle x_{k+1,n}:n\in N_k\rangle$ converges. In this way we recursively construct sets $N_k$ for $k\in\Bbb N$, such that each $N_k$ is infinite, $N_{k+1}\subseteq N_k$ for each $k\in\Bbb N$, and $\langle x_{k,n}:n\in N_k\rangle$ converges for each $k\in\Bbb N$.

Let $m_0=\min N_0$. Given $m_k$ for some $k\in\Bbb N$, let $m_{k+1}=\min\{m\in N_{k+1}:m>m_k\}$, and let $M=\{m_k:k\in\Bbb N\}$. Clearly $M\subseteq\Bbb N$, and by construction $\langle m_k:k\in\Bbb N\rangle$ is strictly increasing, so $M$ is infinite. Moreover, for each $k\in\Bbb N$ we have $m_\ell\in N_\ell\subseteq N_k$ for all $\ell\ge k$.

Now let $k\in\Bbb N$ be arbitrary. Then for each $\ell\ge k$ we have $m_\ell\in N_k$, so $\langle x_{k,m_\ell}:\ell\ge k\rangle$ is a subsequence of the convergent sequence $\langle x_{k,n}:n\in N_k\rangle$ and is therefore itself convergent. But $\langle x_{k,m_\ell}:\ell\ge k\rangle$ is a tail of the sequence $\langle x_{k,m}:m\in M\rangle$, so $\langle x_{k,m}:m\in M\rangle$ converges. That is, $\langle x_{k,m}:m\in M\rangle$ converges for each $k\in\Bbb N$, which is the desired conclusion.

Since my notation is a bit different from yours, it may help if I note that my set $M$ is your $\{j_k:k\in\Bbb N\}$.