Proof. (1): Let

be the left translation action on itself. For each $g \in G$, the functoriality of the Stone-Čech compactification implies that $L_{g}$ extends into a continuous map $\beta L_{g}: \beta G \to \beta G$. By uniqueness of continuous extensions, $\beta L_{g}(\fU) = g\fU$ for all $\fU \in \beta G$. In particular, $\text{Id}_{\beta G}= \beta L_{e}$.

Moreover, for any $g, h \in G$, $\beta L_{gh}= \beta L_{g}\circ \beta L_{h}$, so $g \mapsto L_{g}$ is a homomorphism. Finally, for any $g \in G$,

and $\beta L_{g} \in \text{Aut}(\beta G)$.

Since $\bracsn{L_g1_G|g \in G}= G$ is dense in $\beta G$, $(G \curvearrowright \beta G, 1_{G})$ is an ambit.

(2): For each $g \in G$, let $\pi_{0}(g) = gx_{0}$, then $\pi_{0} \in C(G; X)$, $\pi_{0}(1_{G}) = x_{0}$, and the following diagram commutes

for all $g \in G$. By universal property of $\beta G$, $\pi_{0}$ admits a unique continuous extension $\pi \in C(\beta G; X)$. By uniqueness of continuous extensions, the following diagram commutes

for all $g \in G$. Finally, since $\beta G$ is compact and $X$ is Hausdorff, $\pi(\beta G)$ is also compact and hence closed. As $(X, x_{0})$ is an ambit and $\pi(\beta G) \supset Gx_{0}$, $\pi(\beta G) \supset \ol{Gx_0}= X$.$\square$