# Bolzano-Weierstrauss

This involves the Bolzano-Weierstrauss Theorem, I believe, but I'm not sure where to start.

Prove that the set of open disks in the xy plane with center (x,x) and radius x > 0, x rational, is a countable covering of the set {(x,y): x > 0, y > 0}

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#### Solution Preview

Let A={(x,y),x>0,y>0} and U_x={(u,v):(u-x)^2+(v-x)^2<=x^2, x is a rational number}. So U_x is the set of an open disk with center (x,x) and radius x.

I want to show that {U_x} is a countable covering of A.

First, {U_x} is a countable number of sets. Because the rational numbers is countable and each nonnegative rational ...

#### Solution Summary

This solution is comprised of a detailed explanation to prove that the set of open disks in the xy plane with center (x,x) and radius x > 0, x rational, is a countable covering of the set {(x,y): x > 0, y > 0}.