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(1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute value x + yl of their sum is the sum [x] + [y] of their absolute values. (2) Let x, y and z be real numbers. Show that the distance between a real number x and z is the sum of the distances between x and y and between y and z if and only if y € [x, z]. Illustrate geometrically on the real line. (3) Prove the Generalized Triangle Inequality: if a₁, a2,..., an ER then la₁ + a₂ + ... (Hint: Use the Principle of Mathematical Induction) |≤|a₁|+|a₂|+ ··· ...+lanl. (4) Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B. (5) Show that for any real number x and a subset A of R, exactly one of the following holds: (a) x is an interior point of A, (b) x is a boundary point of A or (c) x is an exterior A and g.l.b. A are in cl A. However, show that (6) Let A be a bounded set of real numbers. Show that each of these need not necessarily be an accumulation point of A. ce or share Corte

(1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute value x + yl of their sum is the sum [x] + [y] of their absolute values. (2) Let x, y and z be real numbers. Show that the distance between a real number x and z is the sum of the distances between x and y and between y and z if and only if y € [x, z]. Illustrate geometrically on the real line. (3) Prove the Generalized Triangle Inequality: if a₁, a2,..., an ER then la₁ + a₂ + ... (Hint: Use the Principle of Mathematical Induction) |≤|a₁|+|a₂|+ ··· ...+lanl. (4) Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B. (5) Show that for any real number x and a subset A of R, exactly one of the following holds: (a) x is an interior point of A, (b) x is a boundary point of A or (c) x is an exterior A and g.l.b. A are in cl A. However, show that (6) Let A be a bounded set of real numbers. Show that each of these need not necessarily be an accumulation point of A. ce or share Corte

Advanced Engineering Mathematics
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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(1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute value x + yl of their sum is the sum [x] + [y] of their absolute values. (2) Let x, y and z be real numbers. Show that the distance between a real number x and z is the sum of the distances between x and y and between y and z if and only if y'e [x, z]. Illustrate geometrically on the real line. (3) Prove the Generalized Triangle Inequality: if a₁, a2,..., an ER then la₁ + a₂ +... ≤la₁l+la₂l++lanl. (Hint: Use the Principle of Mathematical Induction) (4) Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B. (5) Show that for any real number x and a subset A of R, exactly one of the following holds: (a) x is an interior point of A, (b) x is a boundary point of A or (c) x is an exterior point of A. (6) Let A be a bounded set of real numbers. Show that both l.u.b. A and g.l.b. A are in cl A. However, show that each of these need not necessarily be an accumulation point of A. once or shar Corred
Transcribed Image Text:(1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute value x + yl of their sum is the sum [x] + [y] of their absolute values. (2) Let x, y and z be real numbers. Show that the distance between a real number x and z is the sum of the distances between x and y and between y and z if and only if y'e [x, z]. Illustrate geometrically on the real line. (3) Prove the Generalized Triangle Inequality: if a₁, a2,..., an ER then la₁ + a₂ +... ≤la₁l+la₂l++lanl. (Hint: Use the Principle of Mathematical Induction) (4) Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B. (5) Show that for any real number x and a subset A of R, exactly one of the following holds: (a) x is an interior point of A, (b) x is a boundary point of A or (c) x is an exterior point of A. (6) Let A be a bounded set of real numbers. Show that both l.u.b. A and g.l.b. A are in cl A. However, show that each of these need not necessarily be an accumulation point of A. once or shar Corred
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