Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Ans. no. 5 only...
![(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](https://content.bartleby.com/qna-images/question/5453289e-f8e4-46cc-bad8-5270c4260ef2/79c38385-798a-42c6-9fee-cf0267de0dca/8272ot9_processed.jpeg)
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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