Every Positive Number Has A Unique Positive Square Root

**Hello,**

I am currently studying __Introduction to Analysis__ by Maxwell Rosenlicht. On page 28 -29 he gives this proof for Every Positive Number Has A Unique Positive Square Root. Can someone please help me understand a couple of parts of it? I have indicated in bold what I don't understand.

If then . That is, bigger positive numbers have bigger square roots. Thus any given real number can have at most one positive square root.

It remains to show that if , a > 0, then has at least one positive square root. For this purpose consider the set

This set is nonempty, since

, and bounded from above, since if

we have

.Hence, y = l.u.b. S exists.

**I don't understand this part with the max function. Is the point of this to show that for all the elements of S there is a minimum that may or not be in S that is the least upper bound? Also, why is it the max {a,1} and not just x > a?**
We proceed to show that

. First,

, for

, since

. Next, for any

such that

we have

, so

**Also, why is there a min{a,1} here? Why not just drop the min {1,a}?**

since bigger positive numbers have bigger squares. By the definition of

there are numbers greater than

in

, but

. Again, using the fact that bigger positive numbers have bigger squares, we get

Hence

so

The inequality

holds for any

such that

, and by choosing

small enough we can make

less than any preassigned positive number. Thus

is less than any positive number. Since

we must have

, proving

.

**Thank you.**