A sequence of random variables, how to understand it in the convergence theory? I am a bit confused when studying the convergence of random variables. All the material I read using $X_i, i=1:n$ to denote a sequence of random variables. Based on the theory, a random variable is a function mapping the event from the sample space to the real line, in which the outcome is a real value number. Then the $\{X_i(\omega)\}$ is a sequence of real value numbers. There is no confusion here. 
But when talking about convergence of random variables, it goes to $X_n \rightarrow X$ in probability or in distribution. For example, if we toss a coin once, the sample space is $\{tail = 0, head = 1\}$ and the outcome is 0 or 1. If we toss 10 times, each time it is a random variable of outcome 0 or 1. Then no matter how big is the $n$, $X_n$ still equals to 0 or 1 from one tossing. How can we talk about the convergence of random variables from this sense? Unless $X_i$ is the toss of $i=1...n$ times in one experiment with underlying sample space $2^i$, then define a sequence of random variables the number of head counts in $i=1...n$ so that $X_n\rightarrow X$ in probability. 
How should I get my head around this?
 A: As I understand this. You should have some Randome Variables $X_n$ which depends on $n$.
For example, if you take a look at this post:
Convergence in distribution, probability, and 2nd mean
You'll find that if $n \rightarrow \infty$ then $X_n$ converges in probability. Depeding on RVs you have different types of converging. 
For your example you can take $Y_n = \frac{1}{n}\sum_{k=1}^{n}X_k$ and it should converge to 0.5. You could have 10 heads in a row, but as $n \rightarrow \infty$ then $Y_n \rightarrow 0.5$
I hope I answered your question.
A: The following contents are just copy-paste from: Sequence of Random Variables.
Here, we would like to discuss what we precisely mean by a sequence of random variables. Remember that, in any probability model, we have a sample space $S$ and a probability measure $P$. For simplicity, suppose that our sample space consists of a finite number of elements, i.e.,
$$
S=\left\{s_{1}, s_{2}, \cdots, s_{k}\right\}
$$
Then, a random variable $X$ is a mapping that assigns a real number to any of the possible outcomes $s_{i}, i=1,2, \cdots, k .$ Thus, we may write
$$
X\left(s_{i}\right)=x_{i}, \quad \text { for } i=1,2, \cdots, k
$$
When we have a sequence of random variables $X_{1}, X_{2}, X_{3}, \cdots$, it is also useful to remember that we have an underlying sample space $S$. In particular, each $X_{n}$ is a function from $S$ to real numbers. Thus, we may write
$$
X_{n}\left(s_{i}\right)=x_{n i}, \quad \text { for } i=1,2, \cdots, k
$$
In sum, a sequence of random variables is in fact a sequence of functions $X_{n}: S \rightarrow \mathbb{R}$.
