On Page 57 of Jech's Set Theory, Lemma 5.19
If $\kappa$ is a limit cardinal, and $\lambda \geq \operatorname{cf}{\kappa}$, then $\kappa^\lambda = (\lim_{\alpha\to\kappa} \alpha^\lambda)^{\operatorname{cf}{\kappa}}$
I have difficulty in understanding one step of proof, which is
$(\lim_{\alpha\to\kappa} \alpha^\lambda)^{\operatorname{cf}{\kappa}} \leq (\kappa^\lambda)^{\operatorname{cf}{\kappa}}$
Here's how far I understand this inequality. $\lim_{\alpha\to\kappa} \alpha^\lambda = \bigcup_{\alpha<\kappa}\alpha^\lambda$ , the later is the union of function spaces $\{f:\lambda \to \alpha \}$ for all $\alpha < \kappa$. Since $\lambda \ge \operatorname{cf}{\kappa}$, it's possible to have $\sup\{f(\xi): \xi<\lambda\}=\kappa \notin \kappa$. Thus we have an element doesn't belong to $\kappa^\lambda$ ,which implies the reverse $\lim_{\alpha\to\kappa} \alpha^\lambda \ge \kappa^\lambda$ for $\lambda \ge \operatorname{cf}{\kappa}$.