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(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.

(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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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Help me answer number 4 only.
(3) Prove the Generalized Triangle Inequality: if a₁, a2, ..., an € R then [a₁ + a2 + ··· + ªn| ≤ |a₁| + |a₂| + · · · + |an|. (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.
Transcribed Image Text:(3) Prove the Generalized Triangle Inequality: if a₁, a2, ..., an € R then [a₁ + a2 + ··· + ªn| ≤ |a₁| + |a₂| + · · · + |an|. (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.
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