The existence of limit is guaranteed by the (multiplicative) Fekete lemma for the sub-multiplicative sequence $\{\vvvert f^{n}\vvvert\}_{n\in\mathbb{N}}$. The spectral seminorm is sub-multiplicative, and is bounded by the original seminorm $\vvvert\mathord{\cdot}\vvvert$. Moreover, it is power-multiplicative by construction.