I am currently reading a proof in which a fact is used without proof:
For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n \in \mathbb{N}} \| T^n \|^{\frac{1}{n}}. $$
Do we have to show that the sequence $(\| T^n \|^{\frac{1}{n}} )_{n \in \mathbb{N}}$ is decreasing in order to conclude this? Any ideas?
The proof follows from the following theorem from basic calculus:
Let $\{a_n\}_{n\in\mathbb N}$ be a sequence of nonnegative numbers with the property that $a_{n+m}\leq a_n \cdot a_m$ for all $n,m\in\mathbb N$. Then the limit $\lim\limits_{n\to\infty}\sqrt[n]{a_n}$ exists and it is equal to $\inf\limits_{n\in\mathbb N}\sqrt[n]{a_n}.$
Theorem about spectral radius follows after you define $a_n=\|T^n\|$. Note that $\|T^{n+m}\|\leq \|T^n\|\cdot \|T^m\|$ implies $a_{n+m}\leq a_n \cdot a_m.$