Is there a more general definition of a limit? I've been bothered with the standard definition of the limit in analysis for a while: 
'A sequence $(x_n)$ has $x$ as a limit if $\forall \epsilon>0\ \exists N\in\mathbb{N}$ s.t. $\forall n\geq N:\ |x_n-x|<\epsilon$.' 
I feel it is too restrictive. I am wondering if there is some more general definition of a limit, perhaps in terms of a probability measure. Is someone aware of such a kind of limit existing?
This question is mostly inspired by the Bolzano-Weierstrass theorem: 
'Every bounded sequence in $\mathbb{R}$ has a convergent subsequence'. Thus, in every bounded divergent sequence, there is(/are) some hidden limit(s), yet the sequence as a whole is said to have no limit. Could one perhaps define a limit in such a way that these converging subsequences are taken into account?
For example, consider the sequence $(0,1,0,1,0,1,\dots)$. What is the limit of this sequence? I am aware that one could use the Cesaro mean and say that the limit is $\frac{1}{2}$. However, intuitively this doesn't feel right. I'd say the limit is half 0 and half 1. When you consider an element arbitrarily far along in this sequence you lose track of whether you consider an even or an odd element, and you have a probability of $1/2$ of finding a 0 and a probability of $1/2$ of finding 1.
A continuous probability distribution might be a limit as well. I would say the limit of $x\mod 1$ as $x$ tends towards infinity is a uniform probability measure on $[0,1]$.
Is anyone aware of the type of limit I am trying to describe here?
 A: The set of limit points/cluster points/accumulation points of a sequence will usually/always be the set you are trying to describe. Definitions vary depending on the source, but "limits of subsequences" probably captures it unambiguously when you're looking at sequences in the reals (or $\mathbb R^n$).
A: Any particular definition of limit is trying to serve a purpose. The limit definition of a sequence that you mention is the correct one for the purposes of calculus, at least in the sense that it does the job it is designed for. Now, you seem to want a more refined notion of limit of a sequence that will tell you more than just the tail behaviour of the sequence. It sounds like what you are interested in can be achieved as follows. Given a sequence consider the set $L$ of all of its partial limits (let's assume the sequence is bounded, just for simplicity). You then want a probability distribution on $L$ that will have the property: for all $\varepsilon >0$ there exists $N\ge 1$ such that for all $n\ge N$ for all $m\ge 1$, the probability that $|x_k-\ell|<\varepsilon $ for all $N\le k\le N+m$ converges, as $m\to \infty $ to the probability distribution you chose on $L$. If this is what you are interested, then it's a pretty straightforward formalisation. The question is, of course, what is the purpose you have in mind for your generalisation of limit. 
A: One way to make your idea precise is the following: given a sequence $A=(a_i)_{i\in\mathbb{N}}$ (of real numbers, for simplicity), we define the following function $l_A$: for a real number $r$, we let $l_A(r)$ be the supremum of the asymptotic densities of the subsequences of $A$ which converge (in the usual sense) to $r$. So for example:


*

*For the sequence $X=0,1,0,1,...$, every subsequence convering to $0$ has asymptotic density ${1\over 2}$, and similarly for $1$; meanwhile, no subsequence of $X$ converges to anything else. So $l_X(r)={1\over 2}$ if $r\in\{0,1\}$, and $l_X(r)=0$ otherwise.

*The constant sequence $Y=1,1,1,1,...$ has lots of subsequences which converge to $1$, and the supremum of their asymptotic densities is $1$; so $l_Y(r)=1$ if $r=1$ and $l_Y(r)=0$ otherwise.
However, there are two serious problems with this approach.
First, not every subsequence has a well-defined asymptotic density. For example, consider the sequence $Z$ consisting of $2^0$ $0$s, then $2^3$ $1$s, then $2^6$ $0$s, then $2^9$ $1$s, then ... It's easy to check that the subsequence of $0$s has lower density $0$ and upper density $1$, and the same for the subsequence of $1$s. So it's not clear how to define $l_Z$ in this case.
Second, it's not clear why this is interesting. Just because we can make a definition doesn't mean we should. There is certainly something natural about the definition of $l_A$ above, at least for "nice" $A$, but it's not clear to me that there's actually any "meat" here.
