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(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.

(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.

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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Help me answer number 5 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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