Find multiplicatively balanced range Let $0 < x < 1$ and $1 < y < \infty$. Let $X = (x_1,...)$ an equally distributed random variable from the set of all infinite tuples over $[x,y]$ (that is, $[x,y]^\infty$). Let $X_n = x_1\cdot\ldots\cdot x_n$ the product of the first $n$ entries to $X$.
Now if $0 < x < 1$ was given, but $1 < y < \infty$ was variable, I want to find to make $y$ so that $\lim\limits_{n\to \infty} E\left[{X_n}\right] = 1$.
How would I do that?
EDIT: the $\lim$ must be outside to make sense
 A: Nice question (after it was rendered legible). Perhaps somewhat surprisingly, you cannot choose $y$ in this manner.
Multiplying the $x_i$ corresponds to adding their logarithms. By the central limit theorem, the distribution of $\log X_n$ will tend to a Gaussian with mean $n\mu$ and variance $n\sigma^2$, where $\mu$ and $\sigma^2$ are the mean and variance, respectively, of $\log x_i$. There are three possibilities:
If you choose $y$ such that $\mu\lt0$, eventually almost the entire distribution of $\log X_n$ will be concentrated at negative values, so $E[X_n]\lt1$.
If you choose $y$ such that $\mu\gt0$, eventually almost the entire distribution of $\log X_n$ will be concentrated at positive values, so $E[X_n]\gt1$.
If you choose $y$ such that $\mu=0$, the distribution of $\log X_n$ will remain concentrated symmetrically around $0$, but it will spread out and the upper part will blow up $E[X_n]$ while the lower part eventually contributes $0$. Thus $E[X_n]$ will increase without bound.
