Proof that aleph null is the smallest transfinite number? The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page makes the same assertion, again without a proof.
How does one prove there is no smaller transfinite number? Equivalently (I think), why is there no smaller infinite set than the natural numbers?

One property of cardinals says that if there is an injection $f:A \to B$ then the cardinality of $A$ is less than or equal to the cardinality of $B$. If $B$ is infinite, then you can choose any $b_1 \in B$ and define $f(1) = b_1$, and then choose different $b_2 \in B$ so that $f(2) = b_2$, and so forth and you will get an injection $f:{\mathbb N} \to B$ if $B$ is infinite (because for every $n$, you have infinitely many choices left in $B$ for $f(n)$). So $|B| \geq |{\mathbb N}|$ if $B$ is infinite, thus $| \mathbb{N}|$ is the smallest infinite cardinal.
