Question

Prove the following corollary to Dirichlet's theorem: For any irrational α there exist infinitely many distinct rationals p/q such that ∣α− p/q∣ < 1/q²

Answer 1

For any irrational number α, the corollary to Dirichlet's theorem asserts that there exist infinitely many distinct rational numbers
`\( (p)/(q) \)` such that
`\( \left| \alpha - (p)/(q) \right| < (1)/(q^2) \)`. This result follows from the density of the fractional parts of positive integer multiples of α in the unit interval, allowing for the construction of an infinite sequence of rational approximations satisfying the given condition.

Step-by-step explanation:

Dirichlet's theorem states that for any two real numbers
`\( \alpha \)` and β with
`\( 0 < \alpha, \beta \leq 1 \)`, there exist infinitely many positive integers m and n such that
`\( 0 < m\alpha - n\beta < (1)/(√(5)) \)`.

The corollary you provided is a special case of Dirichlet's theorem where
`\( \beta = 0 \)` and
`\( \alpha \)` is an irrational number. The corollary states that for any irrational
`\( \alpha \)`, there exist infinitely many distinct rational numbers
`\( (p)/(q) \)` such that
`\( \left| \alpha - (p)/(q) \right| < (1)/(q^2) \)`.

Let's prove this corollary:

Assume
`\( \alpha \)` is an irrational number. We want to show that there exist infinitely many distinct rational numbers
`\( (p)/(q) \)` satisfying
`\( \left| \alpha - (p)/(q) \right| < (1)/(q^2) \)`.

Consider the sequence of fractional parts of the positive integer multiples of
`\( \alpha \)`:

`\[ \{ n\alpha \} = n\alpha - \lfloor n\alpha \rfloor \]`

Since
`\( \alpha \)` is irrational, the sequence
`\( \{ n\alpha \} \)` is dense in the unit interval
`\([0, 1)\)`. This means that for any small interval (a, b) where
`\( 0 \leq a < b < 1 \)`, there exists a positive integer n such that
`\( \{ n\alpha \} \)` belongs to (a, b).

Now, let's choose a = 0 and
`\( b = (1)/(q) \)`, where q is a positive integer. Since
`\( \{ n\alpha \} \)` can be found in the interval
`\((0, (1)/(q))\)` for infinitely many n, there exist infinitely many positive integers n such that
`\( \{ n\alpha \} < (1)/(q) \)`.

This implies that there exist infinitely many positive integers n and
`\( m = \lfloor n\alpha \rfloor \)` such that
`\( 0 < n\alpha - m < (1)/(q) \)`. Rearranging, we have:

`\[ 0 < \alpha - (m)/(n) < (1)/(qn) \]`

Now, let p = m and q = n. We have shown that for infinitely many positive integers n, there exist corresponding positive integers m such that
`\( \left| \alpha - (m)/(n) \right| < (1)/(qn) \)`.

Since
`\( (1)/(qn) \)` can be made arbitrarily small by choosing large enough n, we conclude that there exist infinitely many distinct rational numbers
`\( (p)/(q) \)` such that
`\( \left| \alpha - (p)/(q) \right| < (1)/(q^2) \)`, proving the corollary.