  We provide a short proof for the convergence of the $n$-th root of $\|A^n\|$
for a complex matrix $A$ by showing $\|A^n\|$ is weakly submultiplicative and
then using Fekete's lemma. This is simpler than the algebraic approach of
Gelfand's formula. Instead of the spectral radius, the limit is expressed as
the infimum of $\sqrt[n]{D\|A^n\|}$ where $D\times D$ is the dimension of $A$
and $\|.\|$ is the maximum norm. A corollary is an effective bound of the
spectral radius. A simple proof using Perron-Frobenius' theorem for the
convergence to the spectral radius for nonnegative matrices is also given. We
address a sibling problem: Given a fixed bilinear map $*:\mathbb C^d\times
\mathbb C^d\to \mathbb C^d$ and a fixed vector $v\in \mathbb C^d$, we prove
that the norm of the sequence $v, v*v, (v*v)*(v*v),
((v*v)*(v*v))*((v*v)*(v*v)), \dots$ is superexponential, that is the $(2^n)$-th
root converges. Bounds for this limit are also given. More complex sequences
are discussed.
