- #1

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## Homework Statement

"Let ##A=\{(x,y)\in ℝ^2: x>0,\space x^2 > y > 0\}##. Prove that for all lines ##b(x)=\langle x,ax \rangle,\space a,x\in ℝ## passing through ##(x,y)=(0,0)##, there exists an interval along that line ##b(x)## that is not in ##A##."

## Homework Equations

Ball of radius ##\epsilon## centered around ##(a,b)##: ##B((a,b),\epsilon)##

Set of points intersected by a line ##l(x)## with length ##|t|⋅|l(x)|##: ##\{t⋅l(x):t\in ℝ\}##

## The Attempt at a Solution

Basically, I constructed an interval of ##b(x)##, with ##|b(x)|=1## centered about ##(x,y)=(0,0)## by forming the intersection: ##B((0,0),\epsilon)\cap \{tb(x):t\in ℝ\}=\{|t|<\epsilon\}##.

I was going to show that if ##t\in (-\epsilon,\epsilon)\cap \{t:xt<0\}\subset \{|t|<\epsilon\}##, that ##\{tb(x)\}⊄A##, but couldn't figure out how to express it. I'm not completely sure that the rough solution is correct, either. For one, the set falls apart whenever ##x=0##. And this is basically the sketch of what I'm working with, here.

There's also a part two to this problem, but I think I should hold back on that until I've finished part one.