## Existence of the Square Root Proof

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?
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 Mentor Blog Entries: 8 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.
 This book never ceases to make me feel stupid. Thanks Micro for making that clearer.

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