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If $b_n$ counts the $n$-step self-avoiding walks from a fixed vertex of a transitive lattice, the connective constant is the exponential growth rateSubmultiplicativity of $b_n$ and the Fekete lemma applied to $\log b_n$ prove existence of this limit.

Splitting any $(m+n)$-step self-avoiding walk after $m$ steps and translating its remaining segment to the origin injects it into an ordered pair of an $m$-step and an $n$-step self-avoiding walk. Hence $b_{m+n}\le b_m{b}_n$. The sequence $a_n=\log b_n$ is subadditive, so the Fekete lemma givesExponentiating proves existence of the connective constant $\kappa ={\lim }_n{b}_n^{1/n}$.