= Consider two metric spaces (X, dx) and (Y, dy). The cartesian product Z X X Y becomes a metric space if equipped with the distance dz((x, y), (u, v)) = dx(x, u) + dy(y, v). (i) Prove that a sequence {(xn, Yn)} converges to (x, y) in Z if and only if Xn → x in X and yn → y in Y.
= Consider two metric spaces (X, dx) and (Y, dy). The cartesian product Z X X Y becomes a metric space if equipped with the distance dz((x, y), (u, v)) = dx(x, u) + dy(y, v). (i) Prove that a sequence {(xn, Yn)} converges to (x, y) in Z if and only if Xn → x in X and yn → y in 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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![**Metric Spaces and Cartesian Products**
Consider two metric spaces \((X, d_X)\) and \((Y, d_Y)\). The Cartesian product \(Z = X \times Y\) becomes a metric space if equipped with the distance
\[
d_Z((x, y), (u, v)) = d_X(x, u) + d_Y(y, v).
\]
**Exercise: Convergence of Sequences**
(i) Prove that a sequence \(\{(x_n, y_n)\}\) converges to \((x, y)\) in \(Z\) if and only if \(x_n \to x\) in \(X\) and \(y_n \to y\) in \(Y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F31dd383a-56c5-43a8-b2d6-6bc509043071%2Fb1gqias_processed.png&w=3840&q=75)
Transcribed Image Text:**Metric Spaces and Cartesian Products**
Consider two metric spaces \((X, d_X)\) and \((Y, d_Y)\). The Cartesian product \(Z = X \times Y\) becomes a metric space if equipped with the distance
\[
d_Z((x, y), (u, v)) = d_X(x, u) + d_Y(y, v).
\]
**Exercise: Convergence of Sequences**
(i) Prove that a sequence \(\{(x_n, y_n)\}\) converges to \((x, y)\) in \(Z\) if and only if \(x_n \to x\) in \(X\) and \(y_n \to y\) in \(Y\).
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