The question is:
Prove that if $\lim\limits_{n\to\infty}X_n = a$, and $X_n > 0$, $n$ is any natural number, then $\lim\limits_{n\to\infty} \sqrt[n]{X_1\cdot X_2\cdot \cdot \cdot X_n} = a$.
I already get stuck on this question for half a day and can only get the following unsatisfactory result:
Consider $\sqrt[n]{X_1\cdot X_2\cdot \cdot \cdot X_n} = a \cdot \sqrt[n]{i_1\cdot i_2\cdot \cdot \cdot i_n}$ where ${i_n}$ converges to 1 as n tends to infinity.
Then I try to prove $\sqrt[n]{i_1\cdot i_2\cdot \cdot \cdot i_n}$ converges to 1 as well. However, utilizing the fact that ${i_n}$ converges to 1 only gets me to $\sqrt[n]{i_1\cdot i_2\cdot \cdot \cdot i_n} <\ (\epsilon + 1)(\epsilon + 1)^{\frac{n-N_0}{n}} \le \epsilon (\epsilon + 2) +1$.
Can anybody please help me understand how should I deal with the question? Thank you.
