Gelfand's formula states that the spectral radius $\rho(A)$ of a square matrix $A$ satisfies
$$\rho(A) = \lim_{n \to \infty} \|A^n\|^{\frac{1}{n}}$$
The standard proof relies on knowing that $\rho(A) < 1$ iff $\lim_{n \rightarrow \infty} A^n = 0$. Is there a proof of Gelfand's formula without using this result, whose proof happens to rely on a burdensome use of Jordan Normal form.