Weakly Compact Cardinal is Strong Limit

Question

A question to Lemma 9.9 in Jech's Set Theory:

Every weakly compact cardinal is inaccessible.

I am working on the part that for $\kappa$ weakly inaccessible, $\kappa$ is strong limit.

Jech writes: That $\kappa$ is a strong limit cardinal follows from Lemma 9.4:
If $\kappa\leq 2^\lambda$ for some $\lambda<\kappa$, then because
$2^\lambda\not\rightarrow (\lambda^+)^2$, we have $\kappa\not\to(\lambda^+)^2$ and hence $\kappa\not\to(\kappa)^2$.

Why do we know then, that there is no $\lambda<\kappa$ with $\kappa\leq 2^\lambda$ and so $\kappa$ strong limit?

I don't see the connection between the partitions and this fact...

I hope someone feels like helping!

Best, Luca

Answer 1

1. If $\kappa=\lambda^+$ then $\kappa\leq2^{\lambda}$ due to Cantor's theorem. Therefore $\kappa$ is a limit cardinal.
2. Similarly, if $\kappa\leq2^\lambda$ for some $\lambda<\kappa$ we have that it's not weakly compact. Therefore $\kappa$ is a strong limit cardinal.