Let $X_1,X_2,...$ be independent Uniform(0,1) distributed random variables.
My first problem ist to show, that $Z_n := \frac{X_1^2+...+Xn^2}{X_1+...+X_n}$ converges almost surely and to find its limit.
My second problem is to show that $\lim_{n\to +\infty} \sqrt[n]{X_1\cdots X_n}=e^{-1}$ almost surely.
Can anybody help me here?
Hint: You can use continuous mapping theorem for a.s. convergence, that is
Assume $f$ is continuous, and $X_n \rightarrow X$ a.s., then $f(X_n) \rightarrow f(X)$ a.s. as well.
Back to your question, in the first question, try to divide both the numerator and denominator $n$, and use apply Law of Large Numbers to $\frac{1}{n} \sum X_n$ and $\frac{1}{n} \sum X_n^2$.
For the second question, consider $e^{\log (X_1 \ldots X_n)^{\frac{1}{n}}} = e^{\frac{1}{n}\sum \log X_n }$.
EDIT
To calculate $E\log X$ it suffices to see that $E \log X = \int_0^1\log xdx = x\log x|^1_0 - \int_0^11dx =-1$. Note that you have to argue $\lim_{x \rightarrow 0} x\log x = 0$ which can be done in many ways, like this.
