Stone-Čech compactification $\beta\mathbb{N}$ of the integers $\mathbb{N}$ with discrete topology has uncountably many points?

Question

How do I show that the Stone-Čech compactification $\beta\mathbb{N}$ of the integers $\mathbb{N}$ with the discrete topology has uncountably many points? There is a hint that crux is to construct a bounded sequence of real numbers which contains uncountably any subsequences converging to different numbers, but I'm still not sure what to do...

Answer 1

Here's an approach using the universal property of the Stone-Čech-compactification:

Let $(q_n)_n$ be an indexing of all rational numbers in the unit interval $I$, and let $f:\Bbb N\to I$ be a map sending $n$ to $q_n$. Then there is a unique map $f':\beta\Bbb N\to I$ such that $f'\iota = f$, where $\iota:\Bbb N\to\beta\Bbb N$ is the canonical embedding. Can you show that $f'$ is surjective?

Answer 2

Let $f: \mathbb{N} \to [0, 1]$ such that the image of $f$ is dense, $\overline{f(\mathbb{N})} = [0, 1]$ (i.e. an enumeration of the rationals).

There exists an extension of $f$ to the Stone–Čech compactification, $\overline{f}: \beta(\mathbb{N}) \to [0, 1]$. The image of $\overline{f}$ is compact and hence closed, yielding$$\overline{f}(\beta(\mathbb{N})) = \overline{f(\mathbb{N})},$$and so $\overline{f}: \beta(\mathbb{N}) \to [0, 1]$ surjects.