# relationship between infinitely often and almost sure convergence

To find a relationship between $P($A_n $i .o)$ and almost convergence we generally use the Borel Cantelli lemma.

But let say $A_n$ = ($X_n \neq$ $Y_n$) where $Y_i= X_i I(|X_i|\leq i)$.

It is given that p( $X_n \neq$ $Y_n$ $i.o$)=$0$.

This implies p( $X_n =$ $Y_n$ $i.o$)=$1$.

My question is this means that $X_n$ - $Y_n$ $\overset{a.s.}{\rightarrow} 0$ ?

Without proving anything , i think this is correct based on my intuition .

Because p( $X_n =$ $Y_n$ $i.o$)=$1$ means no matter how large the n is , $X_n$ equals to $Y_n$. So is this enough to say that $X_n$ - $Y_n$ $\overset{a.s.}{\rightarrow} 0$ in this example ?

• Your conclusion is correct but $X_n \overset {a.s.} \to Y_n$ is not an accepted notation in Mathematics. You should write it as $X_n -Y_n\overset {a.s.} \to 0$ – Kavi Rama Murthy Jun 24 '18 at 12:01
• @KaviRamaMurthy Thank you. I edited the question. – student_R123 Jun 24 '18 at 14:59