Regular cardinals exponentiation.

Question

I am struggling with an exercise regarding cardinal exponentiation when one of the two is regular. The exercise reads:

Let $k$, $\lambda$ be cardinals, with $k$ regular and $\lambda < k$. Prove that $k^{\lambda} = \sum_{\alpha<k}|\alpha|^{\lambda}$ .

I still have to get familiar with cofinalities and regular cardinals so I may be missing something very obvious, but I don't know where to start. Any help is very much appreciated :)

Answer 1

Hints:

1. What do you know about $\operatorname{cf}(\lambda)$ compared to $\operatorname{cf}(\kappa)$?
2. If $f: \lambda \to \kappa$ is a function, what can you say about the range of $f$?
3. Using 2, can you write such $f$ as a function between a different domain and/or codomain?