The following is a theorem:
(Hessenberg) Let $1 \le \kappa \le \lambda$ where $\lambda$ is an infinite cardinal. Then $\kappa \cdot \lambda = \lambda$.
The proof in the book proceeds by transfinite induction showing $\lambda \cdot \lambda \le \lambda$. I have a question about the induction step. In the proof they define a well-order on $\lambda \times \lambda$ and then show that every proper initial segment of $\lambda \times \lambda$ has cardinality less than $\lambda$. It seems long-ish.
Why can't one argue like this: By the inductive assumption we have $\kappa \cdot \kappa \le \kappa$ for all $\kappa < \lambda$. Hence $\sup (\kappa \cdot \kappa) \le \sup \kappa $. But $\sup \kappa = \lambda$ and $\sup \kappa \cdot \kappa = \lambda \cdot \lambda$ which proves the claim.
Thanks for your help.