Please solve as soon as possible 3.Letd(x, y)|2-2for x, y ER.Then is a metric on R. Show that (R, d) is not a complete metric space. (Hint: Consider the sequence(Tn)(Comment: This gives an example of a metric on R with respect to which R is not complete. (Recall that Ris complete with respect to the (usual) Euclidean metric (by Lem. 4.20).))(n), and use the fact that 2n → 0 as n → 00.)

Cauchy sequence: A sequence x1, x2, x3, ... in a metric space (X, d) is called Cauchy if for every…

Answered: 3. Let d(x, y) := |2 - 2| for x, y E R.…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.7: Applied Problems

Problem 58E

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Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric).i. Explain why every horizontal line of the form [a,b] × {y} is compact.ii. Given any open cover Oλ of R, conclude that each line of the form [a,b] × {y} is covered by a finite subfamily of the Oλ's.iii. Using ii, show that R is compact.Hint: In iii, it may help to look at, for each horizontal line, the union of the finite subcover. How "thick" or "thin" can this union be?Note: Using the same argument as that for ℝ, it follows from this problem that a subset of ℝ2 is compact if and only if it is closed and bounded.
consider the metric space < X, d > for the case in which the metric d is the usual metric on R'. Given the closed ball B,(a) C X with centre a = P(3, 1, 1, 1) that is located on its boundary OB. (2,0, 2, 2) and the point (i) Show that every point x 4 Br(a) is the centre of an open ball B:(x) with some feasible radius e > 0, and give the feasible range for ɛ. (ii) Use this to prove that the complement B„(a)° of the close ball is an open set.
Let X=R2 and defined d2: R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a metric on R2
17. Show that the function d is a metric whered(x, y) = |x − y| / (1 + |x + y|)
\(d\) is a metric on \(X\).Show that \(\rho : X^2 \to [0,\infty)\) defined by\[\rho(x,y) = \frac{3d(x,y)}{2 + 3d(x,y)}\]is also a metric on \(X
Suppose a point X lies in the exterior of a disk, and suppose that two lines are drawn from X tangent to the circle. If one line is tangent at Y and the other line is tangent at Z, then show that the line segment XY is congruent to the line segment XZ.
Let (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f is a metric on X.
Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2) Show that p is a metric on R^2.
Let X=R2 and let d be the Euclidean metric. Define d3: R2 x R2 to R by d3(x, y) = min{1, d(x, y)} Verify that d3 is a metric on R2
Can you prove in detail the metric space symmetry condition of the discrete metric, I find myself writing that if d(x,y)=0 and x=y then y=x which means that d(y,x)=0. But I have to prove it in a detailed manner with explanation apparently. Thank you in advance.
If (X, d_{1}) and (Y, d_{2}) are metric spaces, we define d: (X × Y ) × (X × Y ) \rightarrow R by d((x_{1}, y_{ 1}),(x_{2}, y_{2})) = d_{1}(x, x′) + d_{2}(y, y′).Answer the following literals a) Prove that d is a metric b) If X = Y = R with d1 equal to the usual metric and d2 the discrete metric. Describe what the balls are like with the metric d on R^{2}
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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