## Cantor–Schröder–Bernstein Theorem

Knowledge this posts assumes: What is a set, set cardinality, a function, an image of a function and an injective (one-to-one) function.

David Hilbert imagines a hotel with an infinite number of rooms. In this hotel, each room can only be occupied by one guest, and each room is indeed occupied by exactly one guest. What happens if more guests show up? Can they be accommodated for?

PAUSE: WHAT DO YOU THINK AND WHY?

Suppose we propose they cannot be accommodated for, since all the rooms are occupied. Hilbert then claims that he can define the functions $f:A \mapsto B,$ and $g:B \mapsto C,$ where $A$ is a set containing all current guests, and $f$ simply maps each guest to a room in the set $B$, and $g$ maps each room in $B$ to a new one in $C$. Notice that these functions must be injective, since if a room contains two different guests, those two different guests must be the same guest; recall $f(a) = f(b) \rightarrow a = b$.…