Density Property of Real Numbers: Statement, Proof

The density property of real numbers says that between two real numbers there is always an another real number. For example, 1.22 lies between the two real numbers 1.2 and 1.3. In this post, we will state and prove this density property of real numbers.

Statement of Density Property of Real Numbers

Between any two real numbers, there always exists another real number.

Proof of Density Property of Real Numbers

To prove the density property of real numbers, we need to show the following two facts:

1. Let x and y be two real numbers with x < y. Then ∃ a real number r such that x < r < y.
2. For two real numbers x and y with x < y, ∃ an irrational number r such that x < y < r.

As y-x > 0, by the Archimedean property of real numbers, ∃ n ∈ ℕ such that

0 < $\dfrac{1}{n}$ < y-x.

⇒ ny-nx > 1

⇒ nx+1 < ny …(I)

As nx is a positive real number, by this property ∃ an integer m such that

m-1 ≤ nx < m …(II)

⇒ nx+1 ≥ m.

Thus, using (I) we obtain that m ≤ nx+1 < ny. On the other hand, from (II) we have that nx < m. This reduces that

nx < m < ny

⇒ x < m/n < y.

Observe that m/n is a rational number as m is an integer and n is a natural number. Now taking r=m/n, we complete the proof of the first part.

For the second part, note that we claim that √2x and √2y are real numbers. We claim that √2x < √2y.

So by the first part above, ∃ a rational number s such that √2x < s < √2y. This s can be taken non-zero without loss of generality. Now,

x < s/√2 < y.

Take r=s/√2. Then s is an irrational number and x < r < y. This proves the second part.

Combining the above two parts, we deduce that there always exists a real number between any two real numbers. This is the density property of real numbers.