It looks like a typo to me.
For me, the easiest thing to remember is that almost surely if and only if for every we have (the first equality being by definition). Then the result follows from
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?