Relation between spectral radius when the norms are equivalent

Let $|| cdot||_1$ and $|| cdot||_2$ equivalent operator normx on $B(X)$. Hence thereare $a,b >0$ such that

$$a ||A||_1 le ||A||_2 le b||A||_1$$

for all $A in B(X).$ Hence, for $A in B(X)$, we get

$$a^{1/n} ||A||_1^{1/n} le ||A||_2^{1/n} le b^{1/n}||A||_1^{1/n}$$

for all $n in mathbb N.$

With $ n to infty$ we see that the corresponding spectral radii are the same.