How to Show that a Field of Characteristic $0$ is Infinite How can one prove that if a field $K$ has characteristic zero, then $K$ is infinite?

Consider the following sequence of elements
$a_1 = 1$, $a_2=1+1$,..., $a_n=1+1+...+1$ (we sum $n$ times) and so on.
we prove that $\{a_n:n\in\mathbb{N}\}$ is infinite. If by contradiction it's finite then $a_n=a_m$ for some $n<m$ in this case we have $a_m-a_n = a_{m-n}=0$ but then the characteristic is $m-n$ or smaller.
