Let $(x_n)_{n\in\mathbb{Z}_+}$ be a real sequence such that $x_n=1$ for all $n\in\mathbb{Z}_+$.
Consider the sequence $(x_2,x_1,x_3,x_4,x_5,\ldots)$.
Argument 1: $(x_2,x_1,x_3,x_4,x_5,\ldots)$ is a subsequence of $(x_n)_{n\in\mathbb{Z}_+}$, because $$(x_2,x_1,x_3,x_4,x_5,\ldots)=(x_1,x_2,x_3,x_4,x_5,\ldots)=(1,1,1,1,1,\ldots),$$ so that $(x_2,x_1,x_3,x_4,x_5,\ldots)$ is just an ordered infinite tuple that coincides with the original sequence, which is trivially a subsequence of itself.
Argument 2: $(x_2,x_1,x_3,x_4,x_5,\ldots)$ is not a subsequence of $(x_n)_{n\in\mathbb{Z}_+}$. Even though it produces the same infinite tuple as the original sequence (which is undoubtedly a legitimate subsequence), one cannot formally identify $(x_2,x_1,x_3,x_4,x_5,\ldots)$ with $(x_1,x_2,x_3,x_4,x_5,\ldots)$, since the way in which $(x_2,x_1,x_3,x_4,x_5,\ldots)$ was generated involves a rearrangement of indices in such a way that the indices of the new sequence are not strictly increasing.
Which argument do you think is correct? Does the way in which a new sequence is generated from the original one matter in determining whether it qualifies as a legitimate subsequence (Argument 2), or does only the final outcome matter (Argument 1)?
Truth be told, having consulted several formal definitions of subsequences in sundry sources, I am quite confused as to which of these two arguments mathematicians generally accept as the right one. Or is this ambiguity prevalent in the mathematician community? Thank you for sharing your thoughts.