# How do I find the greatest lower bound of a set?

Nov 2, 2014

If the set has the smallest element, then the greatest lower bound of the set is the same as the smallest element; however, if the set does not have the smallest element, then it is a little trickier. Let us look at the following example.

Let

$S = \left\{\frac{1}{1} , \frac{1}{2} , \frac{1}{3} , \frac{1}{4} , \frac{1}{5} , \frac{1}{6} , \ldots\right\}$.

Notice that $S$ does not have the smallest element since ite elements are always smaller and smaller toward zero, but it never quite becomes zero, so all terms are greater than zero, which makes zero a lower bound of $S$. Notice that we can always make the denominator large enough so that the element is smaller than any positive value, which means that no positive value can be a lower bound of $S$. Hence, zero is the greatest lower bound of $S$.

I hope that this was helpful.