# Notation for smallest value greater than a number in a sorted set

In a finite set, is there a concept/notation for the smallest value larger than a particular element? For example, I have a sorted set as $$A = \{a_1, a_2, ..., a_k \}$$ where $$a_2 > a_1, a_3 > a_2, ... , a_{k} > a_{k-1},$$. Given an arbitrary value $$p$$ not be in the set, I want to find the smallest element $$a_k$$ in the set greater than $$p$$. Any idea?

• Not following. Why isn't it just $a_{m+1}$?
– lulu
May 19, 2022 at 16:35
• Maybe the notation you're looking for is $\min\{a\in A:a>x\}$.
– Karl
May 19, 2022 at 16:41
• @lulu Edited the question May 19, 2022 at 16:44
• You use $a_m$ to denote the $m^{th}$ element in the set and also to denote some value which may or may not be in the set? That is extremely poor notation.
– lulu
May 19, 2022 at 16:54
• The term "successor" might be appropriate. Its meaning should be defined in the context, of course. May 19, 2022 at 17:28

As @Karl mentioned in the comments, $$\min\{a\in A:a>x\}$$ might be what you want. Also, for finite sets, you can also use infimum (or $$\inf$$). However, $$\inf$$ only gives the minimum for finite sets and some infinite sets like $$[0,1]$$ - for other infinite sets, it is the greatest lower bound. See this Wikipedia page to learn more: https://en.wikipedia.org/wiki/Infimum_and_supremum
• I was aware of infimum and supremum. However how can they be used with respect to a particular value, such as $p$ described in the question? I think they are absolute for a set, but you can explain in the answer if I am missing anything. May 19, 2022 at 19:54
• @ewr3243 Hello again. How about $\inf\{a\in A:a>x\}$? May 20, 2022 at 11:40