Consider two sequences {$a_n$} and {$b_n$}. The former is defined as {$2^n: n = 0 \text{ to } \infty$} and the latter as { first digit (from the left) of each element in the first sequence}. The first few elements of {$b_n$} are {1,2,4,8,1,3,6,...}. Determine the limit of this ratio R(N): number of 1s up to Nth number / N.
Here is my "solution"
Each element in {$a_n$} can be written in scientific notation as $x_n\times 10^{m_n}$. Denote the first significant digit of $x_n$ as $f_n$, so each element of {$a_n$} is of the form $f_{n}.\#\#\# \times 10^{m_n}$. Now let us consider the sequence {$y_n$}, where each element is defined as $y_n = \log_{10}(x_n)$. Now we see that each $x_n$ is in the interval $[1,10)$ and as a result, we see that each $y_n$ is in the interval $[0,1)$. We claim that the probability density function of $y_n$ is uniform. We will show this by first proving that each element in the sequence {$y_n$} is unique. (Note1: I constructed a contradiction proof which led to the statement $\log_{2}(10) \in \mathbb{Q}$ which is obviously not true).
Each element in {$y_n$} occurs exactly once and each has the same probability of being picked ($=\frac{1}{\infty} = 0$). We can arrange the elements of {$y_n$} in increasing order (call this re-arranged sequence {$Y_n$}). Consider the set $K = \{y \in [0,1)| \exists n \in \mathbb{N}$ such that $y = Y_n\}$. This set $K$ is the set of elements in the sequence {$Y_n$}. Notice that $K$ is a dense subset of $[0,1)$. We see that the normalized probability density function is approximately equal to the box function of height 1 in the interval $[0,1)$. This justifies using the integral in the following steps.
Now, the goal is to find the probability that $f = 1$, i.e. the probability that $1\le x < 2$, i.e. the probability that $0 \le y < \log_{10}(2)$. Hence by simple integration we see $P(f = 1) = \int_{0}^{\log_{10}(2)}1\,dy = \log_{10}(2)$. Hence $P(1) = \lim_{N\to\infty} R(N) = \log_{10}(2)$.
Note2: We can use any base $M$ logarithm as long as $\log_2(M) \notin \mathbb{Q}$. This follows from the change of base formula for logarithms.
My questions: Although I showed that arbitrary logarithms work, how do I show that any function of $x_n$ (call it $y_n(x_n)$) with each element of {$y_n$} unique gives me the same answer? Someone told me to use Liouville's theorem from classical physics but I am not sure how this applies here.
