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Let A CN be a non-empty subset of natural numbers. Prove, using the principle of mathematical induction, that A contains a least element. That is, prove that there exists a natural number n E A such that, for all me A, one has n

Let A CN be a non-empty subset of natural numbers. Prove, using the principle of mathematical induction, that A contains a least element. That is, prove that there exists a natural number n E A such that, for all me A, one has n

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Please write a legible and organized proof by induction. Answer question 1
February 17, 2021 1. Let A CN be a non-empty subset of natural numbers. Prove, using the principle of mathematical induction, that A contains a least element. That is, prove that there exists a natural number n E A such that, for all m E A, one has n <m. 2. Prove that B = {2" | n E N}CR is an unbounded subset of R. That is, prove that there does not exist an r R such that r > 2" for all n e N. 3. Let C CR be a non-empty bounded subset of integers (where bounded means C is bounded above and bounded below). Prove that sup C is an integer.
Transcribed Image Text:February 17, 2021 1. Let A CN be a non-empty subset of natural numbers. Prove, using the principle of mathematical induction, that A contains a least element. That is, prove that there exists a natural number n E A such that, for all m E A, one has n <m. 2. Prove that B = {2" | n E N}CR is an unbounded subset of R. That is, prove that there does not exist an r R such that r > 2" for all n e N. 3. Let C CR be a non-empty bounded subset of integers (where bounded means C is bounded above and bounded below). Prove that sup C is an integer.
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