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 of random variables is said to converge almost surely to the random variable X if , which is equivalent to .

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

is equivalent to

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

is equivalent to ,

which is essentially different statement.

Can anybody comment this?