Does the relative frequency of an event always converge? Suppose that $S$ is the set of possible outcomes for a for a measurable equally likely outcome experiment, and $A$ is a subset of $S$.
Define the relative frequencies $F_A(t)$ to be the sequence of proportions of measured outcome of the experiment is an element of $A$ to the number $t$ of times the experiment carried out.
A standard interpretation of probability (Borel's law of large numbers) states that the probability of the set of outcomes in $A$ is defined to be $\lim_{\;t \to \infty}F_A(t)$. However, there doesn't seem to be any reason that the limit must exist.
1) What proof is given that the sequence of relative frequencies converge?
2) Is there any instances in which this sequence of relative frequencies does not converge?  
 A: You're right that such a limit need not exist:

An example that relative frequencies need not converge:
Consider the following sequence of outcomes:


*

*$X_1\in A$

*$X_k\not\in A$ for $k=2,3$,

*$X_k\in A$ for $k=4,\ldots, 3^2$

*$\ldots$

*$X_k\not\in A$ for $k=3^{n-1}+1,\ldots, 3^n$ if $n$ is odd

*$X_k\in A$ for $k=3^{n-1}+1,\ldots, 3^n$ if $n$ is even

*$\ldots$.


Note that:


*

*if $n$ is even, $F_A(3^n)\geq \frac{3^n-3^{n-1}}{3^n}=\frac{2}{3}$

*if $n$ is odd, $F_A(3^n)\leq \frac{3^{n-1}}{3^n}=\frac{1}{3}$


So $F_A(t)$ does not converge.
(Example from Leon Horsten, personal communication)
There's another example in  Hájek, Alan (2009). Fifteen arguments against hypothetical frequentism. Erkenntnis 70 (2):211 - 235. (http://link.springer.com/article/10.1007%2Fs10670-009-9154-1) Page 220.

But the probability that the sequence of outcomes doesn't converge will be 0. By law of large numbers the probability that the relative frequency converges to the expected value is 1. 
This is a problem for the interpretation of probability that says that the probability is the long-run relative frequency. See, e.g., http://plato.stanford.edu/entries/probability-interpret/#FreInt and the Hájek paper as above for more discussion.
