If $a_n$ is a decreasing series of positive numbers show that $c_n = \sqrt[n]{\prod_{i=1}^n a_i}+ \sqrt[n]{\sum_{i=1}^n a_i^n}$ converges

Question

If $a_n$ is a decreasing series of positive numbers show that $$c_n = \sqrt[n]{\prod_{i=1}^n a_i}+ \sqrt[n]{\sum_{i=1}^n a_i^n}(n \rightarrow \infty)$$

a) converges

b) to a value less than or equal to $2a_1$

I had no trouble showing that the max value at the limit is $2a_1$, but I'm having a little trouble proving that it converges.

Any tips?