I am interested in convergence between countable sequences of real numbers. (Perhaps the definitions to follow are nonstandard. Sorry!)
Say that the sequence $\langle \langle x^1_1,x^1_2,x^1_3,...\rangle, \langle x^2_1,x^2_2,x^2_3,...\rangle, ...\rangle$ pointwise converges to $\langle x_1,x_2,x_3,...\rangle$ iff the sequence $\langle x^1_i, x^2_i, x^3_i,...\rangle$ converges to $x_i$ for all $i$. All $x^j_i$ are real numbers, so the notion of convergence for each "coordinate" is the standard one for real numbers. By a permutation of a set I mean a one-one function from that set to itself.
Given these definitions, is the following statement true?
If $f$ is a permutation of $\mathbb{N}$, and the sequence $\langle \langle x^1_1,x^1_2,x^1_3,...\rangle, \langle x^2_1,x^2_2,x^2_3,...\rangle, ...\rangle$ pointwise converges to $\langle x_1,x_2,x_3,...\rangle$, then also the sequence $\langle \langle x^1_{f(1)},x^1_{f(2)},x^1_{f(3)},...\rangle, \langle x^2_{f(1)},x^2_{f(2)},x^2_{f(3)},...\rangle, ...\rangle$ pointwise converges to $\langle x_{f(1)},x_{f(2)},x_{f(3)},...\rangle$.
What if we make $f$ a finite permutation of $\mathbb{N}$ in the sense that $f$ is a permutation of $\mathbb{N}$ and $f(i)\neq i$ for finitely many $i$ at most?
Any references would be great, too! Thanks!
Your conjectures are certainly true: if $\langle x^1_i, x^2_i, x^3_i,...\rangle$ converges to $x_i$ for every $i$, then $\langle x^1_{f(j)}, x^2_{f(j)}, x^3_{f(j)},...\rangle$ converges to $x_{f(j)}$, because every $f(j)$ equals $i$ for some $i$.
Since it holds for all permutations, it necessarily holds for all finite permutations.
The interesting example to consider is the one where $$x_i^j = \begin{cases} 0, &\text{when $i\ne j$}\\ 10^{10^i}, &\text{when $i= j$} \end{cases} $$ Under your definition, the sequences are converging pointwise to ${\bf 0} =\langle 0,0,0,\ldots\rangle$, while under any reasonable definition of “distance”, each one is much “farther away” from $\bf 0$ than the one before. This shows that pointwise convergence doesn't completely capture our intuitions about covergence in this space.