Density of Q in R

Problem Statement

Prove that for any real numbers $x$ and $y$ with $x < y$ , there exists a rational number $q$ such that

x < q < y.

Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

Solutions

Existence Proof

Since $y-x>0$ , we can choose a positive integer $n$ such that

n(y-x)>1.

Equivalently,

ny-nx>1.

Now let $m$ be the smallest integer greater than $nx$ . Then

m-1 \le nx < m.

So in particular,

nx < m \le nx+1.

But $ny-nx>1$ gives

nx+1<ny.

Hence

nx<m\le nx+1<ny,

nx<m<ny.

Dividing through by $n>0$ ,

x<\frac{m}{n}<y.

Therefore, with $q=\frac{m}{n}$ , we have found a rational number satisfying $x<q<y$ .

\hfill $\square$

Takeaways

Magnification Idea: Multiply the interval by a large integer so its length becomes greater than $1$
Key Step: Once the gap exceeds $1$ , an integer can be placed between the endpoints
Density of $\mathbb{Q}$ : Rational numbers occur between every two distinct real numbers
Big Picture: This is an existence proof rather than a constructive formula for a specific rational
For any real numbers $x$ and $y$ with $x < y$ , there exists infinitely many rational numbers $q$ such that $x < q < y$ .

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
HSC Polynomials: https://vumaths.com/booklets/hsc-polynomials/
HSC Differential Equations: https://vumaths.com/booklets/hsc-differential-equations/

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