Theorem: Let $X_1,X_2,…$ be a martingale with $|X_{n+1}-X_n| \leq M<\infty$. Let $C=\{\lim X_n \text{exists and is finite} \}$ and $D=\{\lim \sup X_n=\infty \text{and} \lim \inf X_n=-\infty$. Then $P(C \cup D)=1$.
Proof: Since $X_n-X_0$ is a martingale, we can wlog suppose that $X_0=0$. Let $0<K<\infty$ and let $N=\inf\{n:X_n \leq -K\}$. $X_{n \wedge N}$ is a martingale with $X_{n \wedge N} \geq -K-M$ a.s. so applying theorem "If $X_n \geq 0$ is a supermartingale then as $n \rightarrow \infty$, $X_n \rightarrow X$ a.s. and $EX \leq EX_0$" to $X_{n \wedge N}+K+M$ shows $\lim X_n$ exists on $\{N=\infty\}$. Letting $K \rightarrow \infty$,we see the limit exists on $\{\lim \inf X_n >-\infty$. Applying the last conclusion to $-X_n$, we see $\lim X_n$ exists on $\{\lim \sup X_n<\infty\}$ and the proof is complete.
This is from Durrett's Probability theory, I don't follow this part of the proof
Letting $K \rightarrow \infty$,we see the limit exists on $\{\lim \inf X_n >-\infty$. Applying the last conclusion to $-X_n$, we see $\lim X_n$ exists on $\{\lim \sup X_n<\infty\}$.
Can someone explain this? Thanks in advance!
Best Answer
Consider the event $\{X_n >-K \, \,\forall n \geq m\}$. Apply the result already proved to the martingale $\{X_m,X_{m+1},X_{m+2},....\}$ to see that $\lim X_n$ exists on this set. Taking union over all $m$ see that $\lim X_n$ exists on $\{\lim \inf X_n>-K\}$. Taking union over $K$ we see that $\lim X_n$ exists on $\{\lim \inf X_n>-\infty\}$.