MathDB

Consider fractions

\frac{a}{b}

where

a

and

b

are positive integers. (a) Prove that for every positive integer

n

, there exists such a fraction

\frac{a}{b}

such that

\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}

and

b \le \sqrt{n}+1

. (b) Show that there are infinitely many positive integers

n

such that no such fraction

\frac{a}{b}

satisfies

\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}

and

b \le \sqrt{n}

A5