= Solution
Write $\mathcal W_n$ for the deterministic <set> of $n$-step <self-avoiding walks> from $0$. Each uses $n$ distinct <edges>, so <independence> gives
$$
\kappa_n=\sum_{w\in\mathcal W_n}\mathbf1_{\{w\text{ open}\}},\qquad\mathbb E_p\kappa_n=c_np^n.
$$
For $n\geq1$, the function $t\mapsto t^{1/n}$ on $[0,\infty)$ is a <concave function>. The <Jensen inequality> gives the <root-moment bound for open self-avoiding walks>
$$
\mathbb E_p(\kappa_n^{1/n})\leq(\mathbb E_p\kappa_n)^{1/n}=p c_n^{1/n}.
$$
Equivalently, the <Lyapunov moment inequality> compares exponents $1/n$ and $1$, both strictly positive. Taking the <limit superior> and the <connective constant> limit proves
$$
\boxed{\limsup_{n\to\infty}\mathbb E_p(\kappa_n^{1/n})\leq p\mu}.
$$
This includes $p=0$ and $n=1$. No <independence> between the different <self-avoiding walk> indicators is asserted or needed.
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