Exercise. Suppose that X is a non-empty set, and that a function p : X x X R satisfies the following two conditions: a) For every x, y E X, we have p(x, y) = 0 if and only if x = y. %3D b) For every x, Y, z E X, we have p(x, y) < p(x, z) + p(y, z). Prove that p is a metric on X. Exercise, Suppose that (an)neN and (yn)nEN are two sequences in a metric space (X, p) such that xn x and yn y as n → 0o. Prove that the real sequence (p(xn, Yn))neN Converges to p(x, y) as n→ 0o. Exercise, a) For every ,x,y E R, let p(x, y) =| x² – y² |. Is pa metric

Exercise. Suppose that X is a non-empty set, and that a function p : X x X R satisfies the following two conditions: a) For every x, y E X, we have p(x, y) = 0 if and only if x = y. %3D b) For every x, Y, z E X, we have p(x, y) < p(x, z) + p(y, z). Prove that p is a metric on X. Exercise, Suppose that (an)neN and (yn)nEN are two sequences in a metric space (X, p) such that xn x and yn y as n → 0o. Prove that the real sequence (p(xn, Yn))neN Converges to p(x, y) as n→ 0o. Exercise, a) For every ,x,y E R, let p(x, y) =| x² – y² |. Is pa metric

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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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

Exercise. Suppose that X is a non-empty set, and that a function p:
X x X → R satisfies the following two conditions:
a) For every x, y E X, we have p(x, y) = 0 if and only if r = y.
b) For every x, Y, z E X, we have p(x, y) < p(x, z) + p(y, z).
Prove that p is a metric on X.
Exercise, Suppose that (xn)nEN and (yn)nɛN are two sequences in a
metric space (X, p) such that xn x and yn y as n → 0. Prove that
the real sequence (p(xn, Yn))neN Converges to p(x, y) as n → 0.
Exercise, a) For every ,x, y E R, let p(x, y) =| x² – y² |. Is pa metric

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