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 $\{X_i\}$ 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?