Convergence of sequences of RV

Question

Let $X,X_n,n\geq 1$ be random variables, and $\epsilon_n >0$. Suppose that: $\sum_{n \geq 1} P(|X_n-X|\geq \epsilon_n)<\infty$

Show that: $X_n \overset{a.s}{\rightarrow}X$, as $n\rightarrow\infty$ when $\epsilon_n\rightarrow 0$.

Answer 1

Denote the sequence of events $(A_n)_n$ as $A_n:=\{|X_n-X|\ge\epsilon_n\}$.

Since we have a convergent series of probability of events, $$ \sum_{n \geq 1} P(|X_n-X|\geq \epsilon_n) = \sum_{n \geq 1} P(A_n) < +\infty $$ it's natural to apply Borel-Cantelli lemma to deduce that $P(\limsup_n A_n)=0$, where $$ \limsup_n A_n := \bigcap_{n\ge1} \bigcup_{k\ge n} A_k = \bigcap_{n\ge1} \bigcup_{k\ge n} \{|X_k-X|\ge\epsilon_k\}. $$ Thus, the event $(\limsup_n A_n)^C$ is realized almost surely. $$ \mathrm{a.s.}\quad \exists {n\ge1}, \forall {k\ge n}, |X_k-X| < \epsilon_k $$ Take $k\to\infty$ and make use of $\epsilon_k \to 0$ to conclude that $$ \mathrm{a.s.}\quad \lim_{k\to\infty} |X_k - X| = 0. $$