Let the set s be given by s = {1/n | n is natural}. describe maximum, minimum, supremum (least upper bound), and infimum (greatest lower bound) for s (they may not exist).

We have that and but the minimum of S doesn't exist.

Step-by-step explanation: To find the supremum (or least upper bound) of a set, we first need to prove that an upper bound exists. So we must find a number such that for all . Note that for every . Thus, for each , we have that . This proves that is an upper bound for . Since the set is bounded above, then we have that the supremum must exist. Notice that which means 1 is the highest number in . So we conclude that .

Now to find the infimum, we must start by finding a lower bound for . This will turn out to be quite easy as we just need to notice that if then . Now we have our lower bound! This means S is bounded below and thus its infimum exists. Let us claim that . As per the definition of infimum, we need to prove that any other lower bound would be smaller than 0. This is not so simple so think of it this way: if we find a number greater than 0 that is also a lower bound for S, then 0 is not the greatest lower bound. Then we will set out to prove that any number greater than 0 is not a lower bound. Let us assume then that . Then there is some such that . Since , then is not a lower bound for . This shows that . Recall that when a set has its infimum as an element, we call it minimum but since we've already proven that every element of S is strictly greater than 0, then the minimum of S doesn't exist.

the answer is d

hope this