Existence of the Square Root Proof

[Ronnin]

I was playing trying to work through a proof in Apostol's Calculus and can't quite understand a step noted. This is from chapter 3, theorem 1.35. Every nonnegative real number has a unique nonnegative square root. The part where you are establishing the set S as nonempty so you can use LUB it is stated that a/(1+a) is in the set S. I've seen different choices for this on other versions of this proof. When I first looked at this I figured it was in S for the reason that that would produce a square of a fraction which would produce something smaller than a. But it looks like this is then used with the binomial theorem to finish off the proof. I don't follow it. Can someone walk me through the logic in this one?

 

[micromass]

Indeed, \frac{a}{1+a} is in S because it is positive and because

\frac{a^2}{(1+a)^2}\leq a

To see this, note that this is equivalent to

a\leq (1+a)^2

or

a\leq 1+a^2+2a

And this is certainly true.

 

[Ronnin]

This book never ceases to make me feel stupid. Thanks Micro for making that clearer.

 

[mathwonk]

do you believe the intermediate value theorem? If so you only need to prove that squaring is continuous. since (a+h)^2 = a^2 + 2h + h^2, it is clear that making h small will make a^2 close to (a+h)^2. qed.

 

[Ronnin]

Now I'm lost again. We are trying to prove that the LUB^2 (LUB=b) cannot be any other value but a. From this point on I don't follow the proof at all. For instance to test if LUB^2>a he sets a number c=b-(b^2-a)/(2b). Where did that come from?

 

[Ronnin]

I know this is binomial trickery but I just don't see it. Any ideas?