Countability of the set $(0,1)$ I am trying to prove that the set $(0,1)$ is uncountable from "A First Course in Analysis by Yau". I have a question about a particular step.
In the text, the result is proved by contradiction. It is supposed that the set $(0,1)$ is countable, which it is then written that there must exist a bijection $f:\mathbb{N}\rightarrow (0,1)$ (which is ultimately contradicted).
My question is, why does the bijective map have to exist? If we suppose that $(0,1)$ is countable, shouldn't there exist an injective map $g:(0,1)\rightarrow\mathbb{N}$?

There is a 1-1 correspondence between $A$ and $B$, if and only if a) there is an injection from $A$ to $B$ and b) at the same time there is an injection from $B$ to $A$.
So, if you can demonstrate that injection from $(0,1)$ to $\mathbb N$, then yes, you have demonstrated that $(0,1)$ is countable -- but more, you have proved it to be '''countably infinite'''.
