Analysis - problem on completeness of the reals

[quasar987]

Homework Statement

Show that for all x\geq 0 in R, there is a real y with y²=x.

The Attempt at a Solution

Let U={u\in R:u^2\leq x}. U is bounded above so let y=sup(U). I'm trying to show that y²=x by showing that the two other alternatives lead to contradictions. First, suppose y²>x. My idea is that I should be able to find a u in U close to y such that x<u²<y². Because y is the supremum of U, for any 0<epsilon<y, there is a u in U with u>y-epsilon <==> u²>(y-epsilon)². Thus, if I find an epsilon<y such that (y-epsilon)²>x, then I will have won. We have (y-epsilon)²>x <==> epsilon² - 2y*epsilon + (y²-x) > 0. If I solve this quadratic inequality for epsilon<y, it gives 0<epsilon<y-\sqrt{x}. So epsilon=(y-\sqrt{x})/2 would do.

But isn't that solution redudant? Because it involves \sqrt{x}, a number that when squared, gives x!

 

[morphism]

Well, you can apply the intermediate value theorem to \sqrt{x} to get a quick proof. If you want to stick to first principles, then I don't think you should be using \sqrt{x} at all.

Let's look at your argument. We want v=y-e with e>0, so v<y, and we want v^2>x. Since we're already assuming that y^2>x, maybe it'd be a good idea to set e=y^2-x. If we try this, we'll find that the y^2 term is annoying. How about we try e=(y^2-x)/y, then? If you try that, you'll see that halving e will be a good idea, i.e. set e=(y^2-x)/(2y).

v^2 = (y-e)^2 = y^2 - (y^2-x) + [(y^2-x)/(2y)]^2 > x

i.e. v is an upperbound for U that is less than y, a contradiction.

(Of course in order to divide by y I had to assume that y>0, i.e. in essence that x>0. But this isn't a problem, because if x=0, then we can deal with this trivial case separately. That is, 0^2=0!)

 

[NateTG]

...
So epsilon=(y-\sqrt{x})/2 would do.

But isn't that solution redudant? Because it involves \sqrt{x}, a number that when squared, gives x!

You can't use \sqrt{x}, but perhaps you can come up with an alternative upper bound for \epsilon that doesn't involve \sqrt{x}?