Basic Analysis I — Real Analysis · Problem 1.2.8

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · density-of-irrationals

Problem

Show that for every pair of real numbers

x

and

y

such that

x < y

, there exists an irrational number

s

such that

x < s < y

. Hint: Apply the density of

\mathbb{Q}

\dfrac{x}{\sqrt{2}}

and

\dfrac{y}{\sqrt{2}}