Let S2 ={x e IR: x > OJ. Does S2 have lower bounds Does S2 have upper bounds Does inf

Step 1 of 2

Let

From the given definition of the set , it can be said that every element in is greater than 0. Therefore, is a lower bound of . Thus, is bounded below.

let us consider another lower bound of .

Suppose . Then, it is obvious that .

Now,

So, . But . It contradicts the fact that is a lower bound of .

Thus, for any other lower bound of , . Thus, every lower bound of is either equal to 0 or smaller than 0. Therefore, 0 is the greatest of all lower bounds of .

Hence, .