A question regarding strong law of large numbers cannot be verified by Borel canteli lemma

Question

Hi my question is this,

let $\mathbb x_1,\ldots,x_n $ be a sequence of Bernouli (p) random variables. Then define

$$\mathbb Y_n=\frac{1}{n}\sum_{i=1}^n x_i$$

Using Strong law of large numbers, $\mathbb Y_n $ converge to p almost surely. I tried to prove this using borel catelli lemma. But i got it as divergent. I have attached my work. Can anyone let me know what i did wrong ?

Thank you.

Answer 1

Hi , This is the suggestion that most of you are telling in the comments section. isnt it ?