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**1. Question I 3.12 from Apostle's calculus volume 1**

If x is an arbitrary real number, prove that there are integers m and n such that m<x<n

If x is an arbitrary real number, prove that there are integers m and n such that m<x<n

**2. Theorem I.27:Every nonempty set S that is bounded below has a greatest lower bound; that is, there is a real number such that L= infS**

**3. Suppose x[itex]\in[/itex]R and belongs to a nonempty set S with no maximum element. It follows that ∃B[itex]\in[/itex]Z[itex]\stackrel{+}{}[/itex] such that B is an upper bound for S. Let n=B, then we have n>x. Similarly, let -S denote the set of negatives of numbers in S. Suppose that -B[itex]\in-Z[/itex][itex]\stackrel{+}{}[/itex] of -S, then by theorem I.27 -B is a lower bound of -S. Let m=-B. , such that m<x<n.**

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