7.2.1 Convergence of a Sequence of Numbers
Before discussing convergence for a sequence of random variables, let us remember what convergence means for a sequence of real numbers. If we have a sequence of real numbers $a_1, a_2, a_3, \cdots$, we can ask whether the sequence converges. For example, the sequence
\begin{align}%\label{} \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \cdots, \frac{n}{n+1}, \cdots \end{align} is defined as \begin{align}%\label{} a_n= \frac{n}{n+1}, \qquad \textrm{ for }n=1,2,3, \cdots \end{align} This sequence converges to $1$. We say that a sequence $a_1$, $a_2$, $a_3$, $\cdots$ converges to a limit $L$ if $a_n$ approaches $L$ as $n$ goes to infinity.
Definition .
A sequence $a_1$, $a_2$, $a_3$, $\cdots$ converges to a limit $L$ if
\begin{align}%\label{} \lim_{n \rightarrow \infty} a_n = L. \end{align} That is, for any $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that \begin{align}%\label{} |a_n-L|<\epsilon, \qquad \textrm{ for all }n > N. \end{align}
