Can you tell me if the following claim and subsequent proof are correct? Thanks.
Claim: If $\alpha = \delta + 1$ is an infinite successor ordinal then $\sum_{\xi < \alpha } \kappa_\xi = \sum_{\xi < \delta} \kappa_\xi$.
Proof: Let $f: \delta + 1 \to \delta $ be a bijection. Then $f$ induces a bijection $F: \bigcup_{\xi < \alpha} \kappa_\xi \times \{\xi \} \to \bigcup_{\xi < \delta} \kappa_\xi \times \{\xi \}$ so that $\sum_{\xi < \alpha} \kappa_\xi = \sum_{\xi < \delta} \kappa_\xi$.
Thanks for your help.