Basic Analysis I — Real Analysis · Problem 1.5.5

Real Numbers — Jiri Lebl

Mathematics — Real Analysis · cardinality

Problem

Using the previous exercise with

b = 2

(binary), show that cardinality of

\mathbb{R}

is the same as the cardinality of

\mathcal{P}(\mathbb{N})

, obtaining yet another (though related) proof that

\mathbb{R}

is uncountable. Hint: Construct two injections, one from

[0, 1]

\mathcal{P}(\mathbb{N})

and one from

\mathcal{P}(\mathbb{N})

[0, 1]

. Hint 2: Given a set

A \subset \mathbb{N}

, let the

n

th binary digit of

x

1

n \in A