Convergence of $V_n=\prod\limits_{i=1}^n U_i$ I struggle to do this exercise:
Let $U_1,U_2,\dots$ be a sequence of i.i.d. random variables. We define
$$V_n=\prod\limits_{i=1}^n U_i$$
Show that $V_n^{1/n}$ converges almost sure and calculate the limit.
So what I thought so far: 
Since I have to show almost sure convergence, I would think that I have to show this with the law of large numbers.
So $$\lim\limits_{n\to \infty}\sum\limits_{i=0}^n\frac{\sqrt{V_n}}{n}=E[\sqrt{V_n}]=nE[\sqrt{U_i}]$$
Now here I'm stucked and I don't know if this is correct. Thanks for help.

Assuming that $U_i > 0$, we may define $X_i$ as $\log U_i$. Assuming that $\mathbb{E}[\log U_i]<+\infty$, the law of large numbers gives
$$ \frac{X_1+\ldots+X_n}{n}\to \mathbb{E}[\log U_i] $$
and by exponentiating back
$$ V_n^{1/n} \to \exp\left(\mathbb{E}[\log U_i]\right). $$
