Cofinality of an infinite cardinal, alternative definition
Here is another approach: assume that $\kappa$ is partitioned by $\mu<\lambda$ many sets $\langle B_\xi\mid \xi<\mu\rangle$, whose cardinality is strictly less than $\kappa$. Especially, we have $$\sup_{\xi<\mu}|B_\xi| = \kappa$$ (since $|\bigcup_{\xi<\mu} B_\xi|=\sup_{\xi<\mu}|B_\xi|$.) It contradicts with the assumption that $\lambda$ is the cofinality of $\kappa$.