another definition of cofinality

Let $\kappa$ be a limit ordinal (e.g. a cardinal). The cofinality of $\kappa$ $\operatorname{cf}(\kappa)$ could also be defined as: $$\operatorname{cf}(\kappa)=\inf \{ |U| : U \subseteq \kappa \text{ such that } \sup \; U = \kappa \} $$ ($\sup \; U$ is calculated using the natural order of the ordinals). The cofinality of a cardinal is always a regular cardinal and hence $\operatorname{cf}(\kappa) = \operatorname{cf}(\operatorname{cf}(\kappa))$ .

This definition is equivalent to the parent definition.