How to prove that $(\frac{a_n}{n})_n$ is convergent in $\mathbb R$?

Question

I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,

Let $E$ be a real Banach space and $T \in \mathcal L (E)$. Let $$ a_n := \log \|T^n\| \quad \forall n \in \mathbb N^*. $$

1. Check that $a_{n+m} \le a_n + a_m$ for all $m, n \ge 1$.
2. Deduce that $\lim_{n \to \infty} \frac{a_n}{n}$ exists and coincides with $\inf_{n \ge 1} \frac{a_n}{n}$. [Hint: Fix an integer $m \geq 1$. Given any integer $n \geq 1$ write $n=m q+r$, where $q$ is the largest integer $\le \frac{n}{m}$ and $0 \leq r<m$. Note that $a_n \leq q a_m+a_r$.]

From the author's hint, I got that $\frac{a_n}{n} \le a_1$ for all $n \ge 1$. I could not even prove that $(\frac{a_n}{n})_n$ is bounded from below. Could you provide me with some more hints (not the full solution) on how to prove that $(\frac{a_n}{n})_n$ is convergent in $\mathbb R$?