random variable convergence
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?