another definition of cofinality

Let $\kappa$ be a limit ordinal (e.g. a cardinal). The cofinality of $\kappa$ $\mathrm{cf}(\kappa )$ could also be defined as:

$$\mathrm{cf}(\kappa )=inf\{|U|:U\subseteq \kappa \text{ such that }supU=\kappa \}$$

($supU$ is calculated using the natural order of the ordinals).The cofinality of a cardinal is always a regular cardinal and hence $\mathrm{cf}(\kappa )=\mathrm{cf}(\mathrm{cf}(\kappa ))$.

This definition is equivalent to the parent definition.