Can a sequence of finite subsets of $\mathbb R$ converge to $\mathbb R$? Assume we have a sequence of finite subsets of $\mathbb R$. Or equivalently a sequence of finite subsets of the closed interval from $0$ to $1$.
Is it possible that this sequence converges to the whole interval? 
Is this an example: take first set to be $\{1/2\}$. Then add points halfway between the end and $1/2$: $\{1/4,1/2,3/4\}$. Continue this. Does this converge to the whole interval?
Consider also the characteristic function for these sets. Do they converge pointwise to constant function $1$ in relation to the standard topology of $\mathbb R$?
 A: To think of a sequence of sets as “converging” to a set is to invite serious and thoroughgoing confusion. (Unless of course you have defined a topology on the set of subsets of your big set (in this case $\mathbb R$ or $[0,1]$). But you haven’t done that.)
Almost surely, you’re really asking about the union of the sets mentioned in the sequence. To decide whether a point is in the union, you must do nothing more than decide whether that point is in one of the sets. Once you realize this, you see that even the point $1/3$ is not in the union.
I want to stress the disastrousness of thinking of infinite union as a kind of convergence: Consider the closed intervals $I_n=[1/n,1-1/n]$. Now the endpoints do indeed converge to $0$ and $1$. But since neither of these two is in any of the closed intervals $I_n$, so neither $0$ nor $1$ is in the union. Two entirely different concepts!
A: Your sequence of sets "converges" to the set of dyadic rationals (rationals with denominator a power of $2$). That set is dense in the interval. So for every real there is a sequence $x_1,x_2,\dots$ converging to that real, with $x_1$ in your first set, $x_2$ in your second set, and so on. 
The characteristic functions converge pointwise to the function that is $1$ on the dyadic rationals and zero elsewhere. 
A: The example of $\{\frac{1}{2}\}$, $\{\frac{1}{4},\frac{1}{2},\frac{3}{4}\}$, ... does not converge to the whole interval. Each added set nests all previous, so just look at the limit of the added sets. It is the set of all binary rational numbers, does not even include all rationals.
The characteristic function similarly does not converge to $1$.
A: There is a sense in which your sets (taken in $[0,1]$) converge to $[0,1]$, though it’s probably not what you had in mind. Let $X=[0,1]$ with the usual topology. For $n\in\Bbb Z^+$ let $$D_n=\left\{\frac{k}{2^n}:0<k<2^n\right\}\;.$$
The sequence $\langle D_n:n\in\Bbb Z^+\rangle$ does indeed converge to $[0,1]$ in the Vietoris topology on the space $K(X)$ of compact subsets of $[0,1]$: if $\langle U_1,\dots,U_n\rangle$ is a basic open nbhd of $[0,1]$ (see the link for notation), then $\bigcup_{k=1}^nU_k=[0,1]$, and it’s clear that for all sufficiently large $m$ we must have $D_m\cap U_k\ne\varnothing$ for $k=1,\dots,n$ and hence $D_m\in\langle U_1,\dots,U_n\rangle$.
A: Assuming that the sequence of finite sets is increasing, $A_i \subseteq A_{n+i}$ :
This is essentially the form of convergence used in defining length of curves, and the Riemann integral.  There limits of finite partitions of an interval are taken with respect to refinement.  Partitions are equivalent to the set of endpoints of the intervals.  
For reasonable increasing sequences of finite sets, there is convergence of the normalized measures that assign equal mass to every point in the set and have total mass $1$.  If the limit of these measures (for intervals) is the length measure on $[0,1]$, or whatever other interval the points were selected from, and the finite sets were formed by adding one point at a time, the sequence of points is called equidistributed. 
When the points are a sequence of independent random samples from a probability distribution, convergence of the uniform measures on the finite samples is the Law of Large Numbers.
When the points are a (generic) trajectory of a dynamical system, the convergence of finite atomic measures is the Ergodic Theorem.
Another commonly used, though somewhat tautological, concept of convergence is that the union of an increasing collection of sets is the direct limit of the collection.  The pointwise limit of characteristic functions of an increasing sequence of finite sets is the characteristic function of the union.  
The concept of a net (vs sequence) is also pertinent.
