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

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 $\sup\{\lambda^+\mid\lambda<\kappa\}=\kappa$.

