- #1

Mr Davis 97

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## Homework Statement

(1) Prove that there exists no smallest positive real number. (2) Does there exist a

smallest positive rational number? (3) Given a real number x, does there exist a

smallest real number y > x?

## Homework Equations

## The Attempt at a Solution

(1) Suppose that ##a## is the smallest real number. Define ##b = \frac{a}{2}##. Then ##0<b<a##, which is a contradiction. Hence, there exists no smallest positive real number.

(2) Since the rationals are a subfield of the reals, the argument above applies here as well.

(3) Let ##x## be an arbitrary positive real number. Suppose that ##y## is the smallest real number such that ##y>x##. Define ##z = x + \frac{y-x}{2}##. Then we have the chain that ##x < z < y##. ##x < z## because ##x < x + \frac{y-x}{2} \rightarrow y > x##, which was assumed to be true, and likewise ##z< y \rightarrow y > x##, which was assumed to be true. In the case that y < x < 0, then just define ##z = -x - \frac{y-x}{2}## and the same result follows. Hence, there does not exist a smallest real number in relation to another real number.