Suppose we have the following random variable:
$X_n = n$ with probability $\frac{1}{n}$ and $0$ with probability $1-\frac{1}{n}$. We we can define this variable on the probability space $([0,1], \mathcal{B}[0,1], \lambda)$ where $\lambda$ is the Lebesgue measure. Also, we have independence for all $n\geq1$.
It is said that this $X_n$ converges to $0$ almost surely (several sources). However, once we check this by the Borel - Cantelli Lemma 2, we get that $\sum_{n=1}^\infty P(X_n=n)=\infty$. Given that the events are independent, we know that $X_n=n$ infinitely often.
What could be the reason I am receiving this contradiction?
This is an answer to the question as posted here. The random variables are not defined explicitly in this question. It is just mentioned that they are defined on $(0,1)$ with Lebesgue measure. Explicit definition was given in the comments after I posted this answer.
I don't know what those sources are but we can only conclude that $X_n \to 0$ in probability. It need not converge almost surely.