# Assuming GCH: if $\mathrm{cf}(\kappa) \leq \lambda < \kappa$, then $\kappa^\lambda = \kappa^+$ (Jech Theorem 5.15)

I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory:

Theorem 5.15 If GCH holds and $$\kappa$$ and $$\lambda$$ are infinite cardinals, then

(i) If $$\kappa \leq \lambda$$, then $$\kappa^\lambda = \lambda^+$$.

(ii) If $$\mathrm{cf}(\kappa) \leq \lambda < \kappa$$, then $$\kappa^\lambda = \kappa^+$$.

(iii) If $$\lambda < \mathrm{cf}(\kappa)$$, then $$\kappa^\lambda = \kappa$$.

In the proof of (ii) he just states that it follows from these two lemmas:

Lemma 5.7 If $$|A| = \kappa \geq \lambda$$, then the set $$[A]^\lambda$$ has cardinality $$\kappa^\lambda.$$

(Here $$[A]^\lambda = \{X \subset A : |X| = \lambda\}.$$)

Lemma 5.8 If $$\lambda$$ is an infinite cardinal and $$\kappa_i > 0$$ for each $$i < \lambda$$, then

$$\sum_{i<\lambda}{\kappa_i} = \lambda\cdot\sup_{i<\lambda}{\kappa_i}.$$

I am struggling to construct an explicit proof using these results and a few facts about cardinal arithmetic such as absorption for infinite cardinals and cardinalities of power sets.

Since, assuming GCH, $$\kappa^+ = 2^\kappa = |\mathcal{P}(A)|$$ for some $$A$$ with cardinality $$\kappa$$, I thought I could come up with some sequence $$\{\mu_i : i < \mathrm{cf}(\kappa)\}$$ such that

$$\kappa^+ = 2^\kappa = |\mathcal{P}(A)| = \sum_{i < \mathrm{cf}(\kappa)} |[A]^{\mu_i}| = \sum_{i < \mathrm{cf}(\kappa)} \kappa^{\mu_i} = \mathrm{cf}(\kappa)\cdot\sup_{i<\mathrm{cf}(\kappa)}\kappa^{\mu_i} = \mathrm{cf}(\kappa)\cdot\kappa^\lambda = \kappa^\lambda$$

Is this approach a good idea? How would I go about finding the appropriate $$\mu_i$$? I suppose it will have to make use of the assumption that $$\mathrm{cf}(\kappa) \leq \lambda < \kappa$$, but I don't see how.

I'd much appreciate any hints on how to proceed (either with my suggestion or another way).

$$\kappa\leq\kappa^\lambda\leq 2^\kappa.$$ Similarly, (5.8) refers to the inequality $$\kappa<\kappa^\lambda\quad\text{ if }\lambda\geq\operatorname{cf}\kappa.$$