random variable convergence

Hello,

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

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

Can anybody comment this?