Why is $\aleph_0$ the smallest cardinal number? It is a well-known fact that $\aleph_0 = \vert \mathbf{N} \vert$ is the smallest infinite cardinal number. But I'm wondering why; does anyone know a proof?
Thanks!

You can show show the following:
Every infinite set $\mathcal{A}$ has a subset $\mathcal{B}$, so $\mathcal{B}\subseteq\mathcal{A}$, which has the same cardinality as $\mathbb{N}$, which is $\aleph_0 = \vert \mathbf{N} \vert=\vert \mathcal{B} \vert$.
This proves that $\mathbb{N}$ has the smallest infinite cardinal number.
