Closure under operations for decreasing sequence

Suppose we have a decreasing infinite sequence a_1 > a_2 > a_3 > a_4 > …. What are some operations its elements are closed under?

Short answer: for a completely arbitrary strictly decreasing infinite sequence $a_1>a_2>a_3>\cdots$ (say in $\mathbb{R}$), you can’t promise closure under the usual arithmetic operations. The only universal “closure” facts are order-based.

What is always closed

• min / max (and all finite order statistics). For any $x,y$ from the sequence, $\min\{x,y\}$ and $\max\{x,y\}$ are just one of $x$ or $y$, hence still in the sequence. More generally, among any finite handful of terms, the $k$-th smallest/largest is again one of those terms.
• “Next element” (successor) map. The map $a_n \mapsto a_{n+1}$ stays inside the set of terms (it’s defined for every $n$). (The “previous element” $a_{n}\mapsto a_{n-1}$ is only partial—undefined at $a_1$.)
• Tail/initial segments. For any $a_n$, the tail $\{a_k:k\ge n\}$ and the initial segment $\{a_k:k\le n\}$ are again made of sequence terms.

You can package min/max as index operations: with $x=a_i, y=a_j$,

$$ \max\{x,y\}=a_{\min\{i,j\}},\qquad \min\{x,y\}=a_{\max\{i,j\}}. $$

What is not guaranteed

• Addition, subtraction, multiplication, division, negation, reciprocals, limits, etc. Counterexamples:

  • Take $a_n=1/n$. Then $a_2+a_3=\tfrac12+\tfrac13=\tfrac56\notin\{1, \tfrac12, \tfrac13,\dots\}$. Also $\inf\{a_n\}=0\notin\{a_n\}$.
  • Take $a_n=-n$. Then $a_2-a_3=-2-(-3)=1\notin\{-1,-2,-3,\dots\}$.

When extra structure does give more closure (special cases)

• Geometric sequences. If $a_n=c r^{\,n-1}$ with $0<r<1$, then products/quotients by powers of $r$ keep you in the set: $a_i a_j = c^2 r^{i+j-2}$ is a constant multiple of some later term; if $c=1$ and the set is $\{r, r^2, r^3,\dots\}$, then $a_i a_j=r^{i+j}$ is exactly another term.
• Other nice forms (e.g., closed under a custom operation defined on indices) are possible, but they depend on the specific sequence.

So, absent further assumptions, the safe universal closures are: min, max, finite order statistics, and the “go to the next term” map.