If $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$

Question

I would like to show that if $2^{\text{cf } \kappa} < \kappa$, then $\kappa^+$ is the least possible value of $\kappa^{\text{cf } \kappa}$ where $\kappa$ is an infinite cardinal. I'm certain it follows easily, but I'm not seeing it. Any hits?

Answer 1

It is well-known (as it follows from König's theorem) that $ \kappa < \kappa^{\text{cf} \ \kappa} $, where $\kappa$ is any infinite cardinal, so your statement is true.

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The above refers to the original question, no longer available.

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To the new question: This statement is also true; see here.