Let $(V, \| \ \|)$ be a Banach algebra. Given two elements $x,y\in V$ satisfying $xy=yx$, prove that $$ \lim_{n\to\infty}\|(x+y)^n\|^{1/n}\le\lim_{n\to\infty}\|x^n\|^{1/n}+\lim_{n\to\infty}\|y^n\|^{1/n} $$ I just do not know where to start. I want to use this inequality to prove a relationship between the spectrum of $x+y, x, y$.
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