I have seen the Shannon capacity defined in two ways:
$\Theta(G) = \sup_k \sqrt[k]{\alpha(G^k)}$
$\Theta(G) = \lim_{k \to \infty} \sqrt[k]{\alpha(G^k)}$
My question is, basically, how do we know the limit (in the bottom one) exists, and how do we know the two definitions are equal?
$G^k$, by the way, stands for the strong product of $k$ copies of $G$.
I do know that $\alpha(G \boxtimes H) \geq \alpha(G) \cdot \alpha(H)$. To see this, let $I_G$, $I_H$ be independent sets in $G$ and $H$. Then, $I_G \times I_H$ is independent in $G \boxtimes H$. At first, I mistakenly decided this meant the sequence $\sqrt[k]{\alpha(G^k)}$ must be nondecreasing. From this, I made sense that the supremum and limit are the same. Since an upper bound is $\chi(\bar{G})$, the limit definition exists. But, this was a bad assumption. I have discovered, using Sage, that
$\alpha(C_5) = 2$
$\sqrt{\alpha(C_5^2)} = \sqrt{5}$
$\sqrt[3]{\alpha(C_5^3)} = \sqrt[3]{10}$
And, $\sqrt{5} > \sqrt[3]{10}$.