
Infimum
I have a hard time understanding the following statement from my textbook (Apostol Calculus Vol.1) :
" Example 1: Let S be the set of all positive real numbers. This set is unbounded above. It has no upper bounds and it has no maximum element.
In example 1, the set of all positive real numbers, the number 0 is the infimum of S. This set has no minimum element."
I understand the first part of the proof about the upper bounds. But why is zero the infimum. Is zero consider a positive real number? Also, in order to be the minimum element, zero has to be included in the set, which I thought it is.

Wikipedia is actually a good place to go for Math help:
"In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all elements of the subset."
So basically, the infimum of the positive real numbers is 0 because 0 is the largest number you can find such that it is less than all the numbers in the positive real numbers.
You can try all you like, but you won't find a bigger number that's not a positive real number and which isn't greater than the greatest element (obviously the positive real numbers don't have a greatest element but this clause is relevant in some other cases).

$\displaystyle \left( {\forall t > 0} \right)\left[ {0 < \frac{t}
{2} < t} \right]$
For any positive number there always one smaller.