GCH implies that $2^{<\kappa}=\kappa$

Question

If GCH holds, then $ 2^{<\kappa}=\kappa$ for all $\kappa$

It is true that $2^{<\kappa}=\sup_{\delta <\kappa}(2^\delta)$

some explain this for me. Thanks in advance

Answer 1

The definition of $2^{<\kappa}$ is $\sup\{2^\lambda\mid\lambda<\kappa\}$.

If $\sf GCH$ holds then $2^\lambda=\lambda^+$.

• If $\kappa=\lambda^+$ then $2^{<\kappa}$ is really just $2^\lambda=\lambda^+ = \kappa$.
• If $\kappa$ is a limit cardinal, then $\lambda<\kappa$ implies $2^\lambda=\lambda^+<\kappa$, therefore $\sup\{\lambda^+\mid\lambda<\kappa\}=\kappa$.