Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent and exponential distributed random variables, where $X_n \sim \mathrm{Exp}(\lambda_n)$ with $\lambda_n = (1-\frac{1}{n+1})^{n+1}$. Let $s<r$ and define:
$$Y_n = \frac{X_1^s + X_2^s + \ldots + X_n^s}{X_1^r + X_2^r + \ldots + X_n^r}$$
Show that $Y_n$ converges almost surely to a limit $Y$ and determines its value.
Firstly, I split up $Y_n$ to $$\frac{X_1^s + X_2^s + \ldots + X_n^s}{n} \cdot \frac{n}{X_1^r + X_2^r + \ldots + X_n^r}$$ and determine the limit for each factor. I should only need the limit for $(X_1 + X_2 + \ldots + X_n)/n$ and apply the continuous mapping theorem for $f_1 = x^s$ and $f_2= x^r$. My problem is to find this limit, cause the law of big numbers doesn't help here cause the RVs are not identically distributed. I hope some can give me a hint on this one.