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I am using the following definition of a stationary set:

Assuming that $cof(\kappa)>\omega$, we call a subset $A$ of $\kappa$ stationary if it meets every closed unbounded subset of $\kappa$.

[Here, for $κ ∈$ Card and $C ⊆ κ$, $C$ is unbounded in $κ$ if $∀α < κ$ $∃β ∈ C$ $α < β$.

$C$ is closed in $κ$ if $∀λ < κ (λ$ is a limit ordinal ∧ $C ∩λ$ is unbounded in $λ→λ ∈ C)$.

Thus $C$ contains its limit points $<κ$.]

Now I need to show that every stationary subset of $\kappa$ has cardinality at least $cof(\kappa)$ and that there is a stationary subset of $\kappa$ of cardinality $cof(\kappa)$

Does anyone have any hint of how to do that?

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For the first one, prove that a stationary set is always unbounded, so its cardinality is at least as $\operatorname{cf}(\kappa)$.

For the second one, prove that if $\alpha=\operatorname{cf}(\kappa)$, then there is a continuous function from $\alpha$ to $\kappa$ with an unbounded range. This range is now closed and unbounded, and therefore stationary.

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    For each $α < κ$, $C_α = κ − α$ is a club. If $S$ is stationary, then $S ∩ C_α \neq ∅$, so that $S$ contains an ordinal bigger than $α$, okay, but I just don't get why $C_{\alpha}$ is a club in $\kappa$ – 2017-02-21
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    Verify the definition. – 2017-02-21