Exercise 3: Proving There Is No Smallest Positive Rational Number — Possible Solution
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THE PROOF
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Claim: There is no smallest positive rational number.

Proof by contradiction. Assume, for contradiction, that a smallest
positive rational number r does exist - meaning r > 0, r is rational,
and no positive rational number is smaller than r.

Consider the number r/2. Since r is rational and 2 is a nonzero
integer, r/2 is also rational (rational numbers are closed under
division by a nonzero integer). Since r > 0, dividing by 2 (a positive
number) preserves positivity, so r/2 > 0 as well. And since r > 0,
dividing it by 2 strictly decreases it: r/2 < r.

So r/2 is a positive rational number, and it is strictly smaller than
r. But r was assumed to be the SMALLEST positive rational number -
meaning nothing positive and rational could be smaller than it. This
is a direct contradiction: r/2 is exactly such a number.

Since the assumption that a smallest positive rational exists leads to
a contradiction, that assumption must be false. Therefore, no smallest
positive rational number exists. QED.

WHY THIS FOLLOWS THIS CHAPTER'S OWN CONTRADICTION STRUCTURE
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Per this chapter, proof by contradiction means "assume the claim is
false, then show this assumption leads to something logically
impossible." The claim here is "there is no smallest positive
rational"; its negation, assumed for contradiction, is "there IS a
smallest positive rational, r." Constructing r/2 directly produces a
positive rational smaller than r - logically impossible given what r
was assumed to be, which is exactly the kind of impossibility this
proof technique is built to expose.

WHY r/2 SPECIFICALLY WAS THE RIGHT NUMBER TO CONSTRUCT
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The proof needed a specific, concrete number that could be shown, with
certainty, to be both (a) rational, (b) positive, and (c) smaller than
r - all three properties needed to be provably true, not just
plausible. Dividing by 2 is a simple, always-valid operation on any
positive rational that guarantees all three properties simultaneously,
which is exactly why it's the standard move for this classic proof.

WHY THIS WORKS AS AN ANSWER
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It follows the contradiction structure precisely - assuming the
negation, constructing a specific concrete counterexample (r/2), and
showing it directly contradicts the assumption's own defining
property - with each step (rational, positive, smaller) justified
individually rather than asserted.