Hello,

question is about about one point in proposition saying that convergence of random variable almost surely implies convergence in probability.

A sequence $\displaystyle (X_n),n\in \mathbb{N}$ of random variables is said to converge almost surely to the random variable X if $\displaystyle P(\omega\mid X_n\to X, n\to \infty)=1$, which is equivalent to $\displaystyle \forall \varepsilon \forall \delta \exists N : P(\bigcap_{n=N}^\infty\[\omega: \mid {X_n - X}\mid< \varepsilon])\geq 1-\delta$.

In proof of this proposition (in book of John B.Thomas) it is said that:

$\displaystyle P(\bigcap_{n=N}^\infty\[\omega: \mid {X_n - X}\mid< \varepsilon])\geq 1-\delta$ is equivalent to $\displaystyle P(\bigcap_{n=N}^\infty\[\omega: \mid {X_n - X}\mid\geq \varepsilon])< \delta$

Although, using De Morgan's law and property of pobability measure I get that

$\displaystyle P(\bigcap_{n=N}^\infty\[\omega: \mid {X_n - X}\mid< \varepsilon])\geq 1-\delta$ is equivalent to $\displaystyle P(\bigcup_{n=N}^\infty\[\omega: \mid {X_n - X}\mid\geq \varepsilon])< \delta$,

which is essentially different statement.

Can anybody comment this?