Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?Bear in mind I can't use facts about continous functions. So this is what I learned in this chapter:Balls/neighborhood centered at a point a consists of all points strictly within some radius r of the point.A set is open if every point in the set has a small ball around it that is contained in the set.A set is closed if its complement is open, and sequentially closed if every sequence in the set has all accumulation points in the set. We proved these are equivalent.A set is sequentially compact if every sequence in the set has accumulation points in the set.A set is compact if every cover by open sets has a finite subcover. Compact sets are sequentially compact.In the reals, compact, sequentially compact, and closed + bounded are equivalent.

Given : As per the question we are given the following bivariate function : x2 + xy + 3y2 = 3 And a…

Answered: Let H be the set of all points (x, y)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let H be the set of all points (x, y) in ℝ² such that x² + xy + 3y² = 3. Show that H is a closed subset of ℝ² (considered with the Euclidean metric). Is H bounded?

Bear in mind I can't use facts about continous functions. So this is what I learned in this chapter:

Balls/neighborhood centered at a point a consists of all points strictly within some radius r of the point.
A set is open if every point in the set has a small ball around it that is contained in the set.
A set is closed if its complement is open, and sequentially closed if every sequence in the set has all accumulation points in the set. We proved these are equivalent.
A set is sequentially compact if every sequence in the set has accumulation points in the set.
A set is compact if every cover by open sets has a finite subcover. Compact sets are sequentially compact.
In the reals, compact, sequentially compact, and closed + bounded are equivalent.

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