Upper and Lower Bounds
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Let's define $S$ as the set of numbers <Sign in to see all the formulas>. This set can have either a finite number of elements or an infinite number of elements.
We define the term bounded to mean that for every <Sign in to see all the formulas>, <Sign in to see all the formulas> where $L$ and $U$ are real numbers. We call $L$ the lower bound and $U$ the upper bound of the set $S$. We can put this in terms of closed intervals and say that <Sign in to see all the formulas>.
With this definition we say that a set is unbounded if for every <Sign in to see all the formulas> there exists an <Sign in to see all the formulas> such that <Sign in to see all the formulas>.
Not all sets have upper and lower bounds. For example, the set of all natural numbers is not bounded above. Some sets may have neither. For example, the set of real numbers is bounded neither above nor below. When a set is not bounded we will use the notation of <Sign in to see all the formulas>. So for every $n_i$ an element of the natural numbers, <Sign in to see all the formulas>. For every $r_i$ an element of the set of real numbers, <Sign in to see all the formulas>.
A set with an upper will have an infinite number of them. If <Sign in to see all the formulas> then isn't <Sign in to see all the formulas>? And also <Sign in to see all the formulas>? Yes! So there are an infinite number of upper bounds. Similarly, if a set has a lower bound then there are an infinite number of lower bounds.
But is there an upper bound that is lower than all other upper bounds. In other words, of all the upper bounds that we have, is there an upper bound that is closest to the set $S$? For real numbers we assume that there is and we call it the least upper bound or the supremum. We denote them by <Sign in to see all the formulas> and <Sign in to see all the formulas>.
The formal definition of the least upper bound is as follows. We say that <Sign in to see all the formulas> is the least upper bound of $S$ if <Sign in to see all the formulas> is an upper bound of $S$ and that for every other upper bound $U$, <Sign in to see all the formulas>.
It is important to note that the least upper bound does not have to be a member of the set. For example, the set $(a,b)$ has the least upper bound of $b$.
Similarly, there is a lower bound greater than all other lower bounds. We call it the greatest lower bound or infimum and denote them by <Sign in to see all the formulas> and <Sign in to see all the formulas>, respectively.
Finally, it is important to mention that for real numbers, we assume that every set of real numbers that is bounded above has a least upper bound. This may be visually obvious, but it is actually an axiom of the real numbers. It oftentimes is called the Least Upper Bound Axiom or the Axiom of Completeness. But what about an axiom for the greatest lower bound? Given the properties of the real numbers and the axiom of completeness, we can prove that there is a greatest lower bound.
