I am new in random processes and in one of my homework questions the Porfessor asks the following:
Prove that if $\lim_{n\to\infty} P\{\sup_{k>n} |X_k-m|>= \varepsilon\} = 0$ the sequence of random variables $(X_n)_{n\in\mathbb{N}}$ converges almost surely to the constant $m$.
Any help?
thank you
Let $\varepsilon>0$. We have $P(\bigcap_{n\in\mathbb{N}}\{\sup_{k>n}|X_k-m|>\varepsilon\})=\lim_{n\rightarrow\infty}P(\sup_{k>n}|X_k-m|>\varepsilon)=0$ and it follows that $\lim_{n\rightarrow\infty}|X_n-m|\leq\varepsilon$ a.s. Since $\varepsilon$ was arbitrary, the result follows.