Does a Smallest Real Number Exist for a Given Real Number?

[Mr Davis 97]

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.

 

[andrewkirk]

Looks good.

(3) could be made shorter by using the result from (1). Just show that if y is the smallest real number > x then (y-x) is the smallest positive real number.