Convergence of power of a convergent sequence?

Question

Let $\{a_k\}$ be a monotonically decreasing sequence of positive real numbers such that $a_k \to a$. Then can we say that $a_k^p \to a^p$ for any positive real $p > 1$.

Answer 1

Yes, since the functions $x\mapsto x^p$ is continuous. And the fact that the sequence is monotonic doesn't matter.

Answer 2

Yes, the limit of a product is the product of the limits, provided that each individual factor's limit exists, which you are given as an assumption.