Basic Analysis I — Real Analysis · Problem 1.5.3

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · decimal-representation

Problem

Show that real numbers

x \in (0, 1)

with nonunique decimal representation are exactly the rational numbers that can be written as

\dfrac{m}{10^n}

for some integers

m

and

n

. In this case show that there exist exactly two representations of

x