# Ambiguity of Defining Subsequences

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.

• Only the final outcome matters, i.e. Argument 1 is correct. One would say (most of) today’s mathematics gives the “denotation” priority over the “sense”. The book “Proofs and Types” by J. Girard is available online and the beginning of it may interest you because it treats some related themes. Oct 15 '13 at 12:58
• @EwanDelanoy. Thank you for the clarification. I'm reading the real-analysis book of Folland (1999), which, AFAIK, is quite an authoritative source. In Exercise 4.31 (p. 127), he claims that if $(x_n)_{n\in\mathbb N}$ is a sequence and $k\mapsto n_k$ is a map from naturals to naturals, then $(x_{n_k})_{k\in\mathbb N}$ is a subsequence iff $n_k$ is strictly increasing. The “if” part is clear. But the “only if” follows only if (no pun intended) you stick with Argument 2. That is, Folland seems to implicitly include the map $k\mapsto n_k$ in the underlying definition of a subsequence. Oct 15 '13 at 15:12
• Continued: In other words, authors according to whom Argument 1 is correct may just merely require the existence of a strictly increasing index-subsequence generating a given subcollection of the original sequence to be a legit subsequence, while those advocating Argument 2 deem the mapping generating the subsequence of indices an integral part of the definition of a subsequence, even in cases when the final outcome is the same with another, strictly increasing index subsequence. It's quite a subtle point, but it may give rise to great confusion... Oct 15 '13 at 15:16

For this particular sequence, $(x_2, x_1, \ldots)$ is a subsequence. Your second argument states correctly, that in general $(x_2, x_1, \ldots)$ is not a subsequence of $(x_n)$, or otherwise: There is a sequence $(y_n)$ (necessarily different from $(x_n)$) such that $(y_2, y_1, \ldots)$ is not a subsequence of $(y_n)$, an example being $(y_n) = (n)$.