Limit of sequence of sequences

Question

I got to thinking about sequences of Cauchy sequences. Here is a simple example. Let us define $b_n = (n,1,1,1,...)$ for $n\in\mathbb N$. So we have \begin{align} b_1&=(1,1,1,1,...)\\ b_2&=(2,1,1,1,...)\\ b_3&=(3,1,1,1,...)\\ \dots \end{align} Question: what is $\lim_{n\to\infty} b_n$? It looks like that is not a valid sequence. Does this question even make sense? (I am not sure how such a limit would even be defined.)

Answer 1

A sequence in $X$ is actually a function $\mathbb N\to X$ or equivalently an element of the set $X^\mathbb N$.

If $X$ is a topological space then $X^{\mathbb N}$ can be equipped with a topology that corresponds in some way with the original topology on $X$, which gives birth to the possibility of convergence of sequences.

There are several candidates for the topology on $X^{\mathbb N}$.

In each of them a sequence of sequences might have a limit.

The question whether $\lim_{n\to\infty}b_n$ exists (so that the expression makes sense) depends heavily on this context.