# Does a sequence of Bernoulli random variables with parameter $1/n$ converge to $0$ almost surely?

Consider independent random variables $$X_n\sim\operatorname{\mathsf{Ber}}\left(\frac1n\right)$$. The problem is whether $$X_n\xrightarrow{\mathrm{a.s.}}0$$ is true or not. I tried to use the definition $$\Pr\left(\lim_{n\to\infty}X_n=0\right)=1$$ But I do not understand what the limit of those mean. I read that it can be rewritten like this $$\Pr\left(\omega:\lim_{n\to\infty}X_n(\omega)=0\right)=1$$ and it does not help because I am not aware of what $$\Omega$$ is. So, what does the limit above mean and how do we find the $$\Omega$$ like this for given $$X_n$$? For context, I have seen this question, but I do not know Borel-Cantelli or Lebesgue theory.

My friend and I were discussing and he suggested the following. Let $$\Omega$$ be the set of all sequences $$(x_1,x_2,\dots)$$ with each $$x_i$$ chosen uniformly from $$[0,1]$$ (that is the probability assignment). Then, define $$X_n$$ to be the indicator of $$x_i\leq\frac1n$$. Then, for the sequences of the form $$x_k=1/k^{\alpha}$$ for some $$\alpha>1$$, we have that $$X_n$$ does not converge to $$0$$. However, we do not know if this set has nonzero probability, which would imply that the almost sure convergence is not true.

• Try using the second Borel-Cantelli lemma, noting that $\sum_{n=1}^{\infty}1/n = \infty$ along the way... Nov 17, 2023 at 18:09
• @jbowman: I don't understand how to use it here. What are the $E_n$s? Is there a proof or intuition directly based on the definition of a.s. convergence? Nov 18, 2023 at 14:12
• $E_n$ is the $n^{th}$ event whose probability you are looking at, e.g., $X_n = 1$. 2BC tells us that $X_n = 1$ will occur infinitely often. See if you can relate that to the definition of a.s. convergence. Nov 18, 2023 at 15:30