I came across this property while studying material on Reimann Integral, i.e when is a function integrable. (When $inf{U} = sup{L}$, where U and L are the upper and lower bound of sums).
For a set S, corresponding to any $\epsilon_1>0$, $\exists$ an element of the set S (say $X_i$), such that
$X_i$ < $inf{S}$ + $\epsilon_1$
Similarly,
$X_j$ > $sup{S}$ - $\epsilon_2$
Here, $infS$ is the greatest lower bound of the set, so I can see how adding some constant to this lower bound can exceed some other element of the set. But how can I realize that this will be true for all the elements of the set. Also, I don't understand the utility of this result.
Thanks for helping out.