Past exam of the logic and set theory course of the University of Oxford / 2008 / B / 7 / 3

There is an injection from the power set of $\omega$ into the real numbers given byIf $S$ and $T$ first differ at $k$, the contribution at $k$ has magnitude $2/3^{k+1}$, while the entire later tail has magnitude at mostTheir real values are therefore different. This proves injectivity directly and avoids any ambiguity of numeral expansions.

Conversely, map a real number $x$ to the set $\{q\in \mathbb{Q}:q<x\}$. Distinct real numbers give distinct sets because the rational numbers are dense in the real line. Fixing a bijection between $\omega$ and the countable set $\mathbb{Q}$ turns this into an injection $\mathbb{R}\to \mathcal{P}(\omega )$. Cantor-Schröder-Bernstein theorem now gives