Any nonempty subset of Reals(R) which is bounded above must have a least upper bound (aka...

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Any nonempty subset of Reals(R) which is bounded above must have a least upper bound (aka Supremum) Archimedean Property-Suppose a,b in R with a,b>0. Then there exists an n in N (natural numbers) where na> b. Denseness of Q - Suppose a, b in R with a<b. Then there exists an r in Q(rational numbers) such that a<r<b. Lemma: Suppose a, b in R with b-a>l. Then there is an integer n in (a,b). Using only the properties above, prove the problem below. Prove that if a>0, then there exists an n in N such that 1/n < a <n. Any nonempty subset of Reals(R) which is bounded above must have a least upper bound (aka Supremum) Archimedean Property-Suppose a,b in R with a,b>0. Then there exists an n in N (natural numbers) where na> b. Denseness of Q - Suppose a, b in R with a<b. Then there exists an r in Q(rational numbers) such that a<r<b. Lemma: Suppose a, b in R with b-a>l. Then there is an integer n in (a,b). Using only the properties above, prove the problem below. Prove that if a>0, then there exists an n in N such that 1/n < a <n.

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The proof is given below By the Archimedean proper... View the full answer