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How to show that in an uncountable set of real numbers, for all $x,y$, there exists $z$ such that $z^2=x+y$
Show that in an uncountable set of real numbers, for all $x,y$, there exists $z$ such that $z^2=x+y$.
I’m struggling with this problem.
I’m not sure of my approach in solving this problem. For each $x,y$, I would take a real number $z$, and I would be able to show that:
$zfrac{y}{2}$ and $z^2-z(x+y)>0$
However, this is the part of the proof that I am unsure of. How can I show the two inequalities?
How do I know that I should not use $epsilon$ and $delta$? I feel like this should have an answer somewhere, but I can’t seem to find it.
A:
You seem to be getting close. Here’s the proof of the two inequalities:
For $z$ to be positive, we need $zgeqslant0$; and if $zgeqslant0$, then $zgeqslantdfrac{x}2$ and $zgeqslantdfrac{y}2$.
If $z
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TIPPECANOE, IN. — It’s packed inside – some 50 students and faculty, plus a couple of local politicians, and a lot of the phone booths in town are literally loaded with students begging to be shot.
You do the math.
These students aren’t just curious, they’re pumped up and ready to shoot. I sat down with them at Columbus High School one day last week to find out why they were so excited about this much-hyped rite of passage.
Some of them didn’t even wait until after fifth period to ask — instead, they cut class, called their parents and signed up as soon as they got home.
They weren’t the only ones thinking this would be a great way to leave school behind.
In the parking lot of that school on a snowy Monday morning, there was a line of 15 or 20 pickup trucks, and – believe it or not – the steering wheels of four or five of them had been cut off.
Word was spreading. Two days earlier, a Columbian student had shown up at the county clerk’s office and had his attorney put in a change-of-address card, saying he was moving to Tippecanoe and wanted to be known as “The Watford County Shooter.”
Police departments are monitoring local gun shops and issuing bulletins, and local merchants say that gun sales in this community have doubled in the past month.
“That’s pretty unusual for this time of year,” said Tom Merrel, who runs the Shooting and Hunting Supply store on County Road 166, just a few minutes from the high school. “I haven’t seen this much interest in a gun show since the last gun show.”
Asked if he thought kids were getting their scopes and scabbards from the wrong side of the Ozarks, Merrel
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