Cardinality of a club

Question

I proved that a club $C$ in $\kappa$ has the same cardinality as $\kappa$. Is it really true ? Thanks.

Answer 1

Every club is an unbounded set. If $\kappa$ is regular this means that every unbounded set has order type $\kappa$ and therefore of size $\kappa$.

If $\kappa$ is singular then it is not true, take $\{\aleph_\alpha\mid\alpha<\omega_1\}$ as a club of $\aleph_{\omega_1}$.