I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum (or a greatest lower bound) for some set $S$.
How many suprema and infima can a set have that he has used determinents like a and an instead of the.
As far as my learning is concerned, I was sure about the fact that supremum and infimum are unique for a given set.
On
In general, you can have more than one infimum. Consider sets of pairs of numbers $(a,b)$ and the relation on pairs $(a,b) \succeq (c,d)$ iff $a\ge c$ and $b\ge d$. This is only a partial order, and so, with respect to this relation, the following set has two distinct infima: $\{(1,2),(2,1),(2,2)\}$. That is, there are two elements that satisfy: "no element of the set is less that this element".
If the set is totally ordered (like the reals) then the infimum and supremum are unique.
Let $(X,\leq)$ be a partially ordered set. That means that for all $x,y,z\in X$:
1) $x\leq x$ (reflexivity)
2) $x\leq y$ and $y\leq x$ imply $x=y$ (anti-symmetry)
3) $x\leq y$ and $y\leq z$ imply $x\leq z$ (transitivity)
An element $x\in X$ is a lower bound of a subset $S\subseteq X$ if $x\leq S$ for all $s\in S$. An element $x$ is the greatest element in a subset $T\subseteq X$ if $x\in T$ and $t\leq x$ for all $t\in T$. An infimum of a set $S$ is a greatest element in the set of lower bounds of $S$. We will show that there can be at most one greatest element in every set, so there can be at most one infimum for every set.
There can be at most one greatest element in a subset $T\subseteq X$.
Proof: Let $x,x'$ be both greatest elements in $T$. Then $x\in T$ and $x'\in T$ and since $x$ is a greatest element in $T$, we have $x'\leq x$. Similarly, $x\leq x'$. By anti-symmetry, $x=x'$.