Say, $a_n → \alpha$. ($\alpha \in \mathbb{R}$)
How do I prove that the sequence $\{a_n ^r\}$ converges to $\alpha ^r$ when $r\in \mathbb{R}$? (Only if $\alpha ^r$ can be defined)
Say, $a_n → \alpha$. ($\alpha \in \mathbb{R}$)
How do I prove that the sequence $\{a_n ^r\}$ converges to $\alpha ^r$ when $r\in \mathbb{R}$? (Only if $\alpha ^r$ can be defined)
For $r\in\mathbb Q$, you should consider 3 cases:
For arbitrary $r\in\mathbb R$, in a sense you have to use continuity, anyway exactly continuity defines $\alpha^r$ for all $r\in\mathbb R$..