Intersections of fewer than the cofinality many clubs

For $\mu =0$, the intersection is $\theta$, which is closed and unbounded in itself. For $\mu >0$, let $f_i(\beta )=\min (C_i\setminus (\beta +1))$ and $g(\beta )={\sup }_{i<\mu }{f}_i(\beta )$. The supremum is below $\theta$, and $g$ is nondecreasing and strictly above its argument.

The nonzero closure points $\delta$ of $g$ are unbounded by the closure lemma. For every $i$ and $\beta <\delta$, $\beta <f_i(\beta )\le g(\beta )<\delta$. Thus $C_i\cap \delta$ is unbounded in $\delta$; $\delta$ is a nonzero limit and belongs to each $C_i$. This proves unboundedness of the intersection.

If $\delta$ is a nonzero limit point of the intersection, it is a limit point of each $C_i$, hence lies in each. This proves closure; the case $\mu =1$ is included. Specializing $\theta =\kappa$ gives the last assertion.