I can't find the mistake in this argument (Cofinality and König's theorem) I have some trouble explaining this apparent contradiction: we know that given $k>\aleph_0$ an infinite cardinal, $\mu=cof(k)$ is the minimum cardinal such that $k=\sum_{i\in \mu} k_i$ where $|k_i|<k$. But, using König's theorem $\sum_{i\in \mu} k_i< \prod_{i\in \mu} k=k\mu=k$ and apparently we get $k<k$. Where is my mistake? Thank you!

Your mistake is that $$\prod_\mu\kappa=\kappa^\mu$$ And the result of the theorem is that it is indeed larger than $\kappa$