Rado's function $\Sigma (n)$ gives us the maximum number of 1's that a Turing machine may write on an initially blank tape. It is widely known that this function is non-computable by a Turing machine, but I am struggling to find a proof of this.
I have found one proof (here), but I'm not entirely sure that I understand the idea behind it, which states that
We can show that Rado's function $\Sigma (n)$ is non-computable by showing that if $f(n)$ is any computable function, then there exists $n_0$ such that $\Sigma (n) > f(n)$ for all $n \geq n_0$.
Why does showing this (which I assume is equivalent to showing that Rado's function grows more quickly than any other computable function) prove that Rado's function is non-computable?
Alternatively, how else might we prove the non-computability of Rado's function?