I'm doing a lot of practice problems with sequences, and I've noticed a number of problems ask about the convergence of the sequence raised to a positive power. It seems like in all the examples that I've tried, if $a_n$ converges to $a$, then $a_n^c$ converges to $a^c$, where $c$ is some positive real number. Is this always true?
I want to say yes, since we can define a new sequence $b_n$ as the product of $a_n$ and use the Algebraic Limit Theorem, but I'm wondering if there are any special cases I'm failing to consider.