Monotone Sequences and Subsequences: Comparison of All Terms, Growth of the Indices, and Subsequences of Subsequences

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let monotone sequences and subsequences be as in Monotone Sequences §monotone and Subsequences §subsequence, with $\mathbb{N}$ carrying its order $\le$ of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §order, a total order with strict relation $<$ by Arithmetic and Order of the Natural Numbers §partial-order and Arithmetic and Order of the Natural Numbers §trichotomy.

¶ Let $(a_{n})$ be a sequence in a set $X$ with a partial order $\le$ and strict relation $<$, and let $m,n\in\mathbb{N}$ with $m\le n$. If $(a_{n})$ is nondecreasing, then $a_{m}\le a_{n}$; if it is nonincreasing, then $a_{n}\le a_{m}$. If moreover $m<n$ and $(a_{n})$ is strictly increasing, respectively strictly decreasing, then $a_{m}<a_{n}$, respectively $a_{n}<a_{m}$.

Let $\sigma,\tau:\mathbb{N}\to\mathbb{N}$ be strictly increasing.

¶ $k\le\sigma(k)$ for every $k\in\mathbb{N}$, and $\sigma(j)<\sigma(k)$ if and only if $j<k$, for all $j,k\in\mathbb{N}$.

¶ $\sigma\circ\tau$ is strictly increasing, and $(a\circ\sigma)\circ\tau=a\circ(\sigma\circ\tau)$ for every sequence $a$ in a set; hence a subsequence of a subsequence of a sequence is a subsequence of it.