I read the Borel-Cantelli lemma, but I am not that familiar with probability. I would like to see this lemma with an illustration of a simple example.
I am not too interested to see the proof of this lemma, but for my understanding, I would like to see this through examples. Whilst searching online for applications of the lemma, I couldn't find any article in Amer. Math. Monthly, of Mathematics Gazette, or other, but only research papers, which are hard to read for me 'now'. Also, while studying limit superior and limit inferior defined for sequence of sets, the examples usually involved are coming from the Borel-Cantelli lemma. I will be happy if one explains this lemma by a simple example.
Informally, the Borel-Cantelli lemma states that if the sum of the probability of sequence of events $\mathbb{E}_1,\cdots ,\mathbb{E}_n$ have finite sum, then the probability that infinitely many of them occur is zero. There are plenty of examples, but here is one; let $\{ X_n \}$ be a sequence of random variables on the interval $(0,1)$. Let $\mathbb{E}$ denote the expectation. We can show that $X_n\to X$ almost surely whenever \begin{equation*} \sum_n \mathbb{E}(|X_n-X|^r)<\infty \end{equation*} holds for $r>0$. By an elementary application of Markov's inequality, we have \begin{equation*} \sum_n \mathbb{P}(|X_n-X|>\epsilon)\leq \sum_n \frac{\mathbb{E}|X_n-X|^r}{\epsilon^r}<\infty,~\epsilon>0. \end{equation*} The result follows from the Borel-Cantelli lemma.
If you want to brush up on your probability, try studying the following:
Schaum's Outline of Probability and Statistics by Murray Spiegel
A First Course in Probability by Sheldon Ross