Limit points of an unbounded set form a club

Define $f(\beta )=\min \{x\in X:x>\beta \}$. This exists by unboundedness, is nondecreasing, and satisfies $f(\beta )>\beta$. Its closure points form a club. A nonzero closure point cannot be a successor $\gamma +1$, since $f(\gamma )\ge \gamma +1$; and for every $\beta <\delta$ a nonzero closure point has $\beta <f(\beta )<\delta$. Thus it is in $\mathrm{acc}(X)$. Conversely each nonzero limit point of $X$ is closed under $f$.

Removing zero from the closure-point club preserves unboundedness and closure at nonzero limits. Equivalently, closure of acc follows directly: below a limit of limit points, first choose a limit point above a given bound, then a point of $X$ above that bound. Hence acc is club with exactly the stipulated nonzero-limit convention.