Gimel function and cofinality.

Question

I am supposed to show that under the conditions given below, the next property is true:

Let $\lambda \in Card$, where $Card$ is the class of all infinite cardinals. Then,

If $\kappa>\lambda$ and $\forall\eta<\kappa$ $\eta^{\lambda}<\kappa$ and if cof$(\kappa)\leq\lambda$, then $\kappa^{\lambda}=\kappa^{cof(\kappa)}$

Here $cof(\kappa)$ is the standard notation for cofinality in context of the minimal cardinality of all cofinal subsets of $\kappa$.

One inequality is obvious, i.e. $\kappa^{cof(\kappa)}\leq\kappa^{\lambda}$ because of one of the assumptions above, but I am struggling with the other one.

Answer 1

Hint: If $S\subseteq\kappa$ is unbounded, then $\kappa^\lambda\leq\prod_{\alpha\in S}|\alpha+1|^\lambda$.

If you can't figure out why this inequality is true, more details are hidden below.

A function $f:\lambda\to\kappa$ is determined by its restriction to partial functions $\lambda\to \alpha$ for each $\alpha$ and such a partial function can be thought of as a total function $\lambda\to\alpha+1$ (send the undefined values to $\alpha$).