infinitisimal and infinity

Hi,

I have a question. Let's say an experiment has a very little probability of giving the result A and that probability is either (treat every case separately) :

1) it is a very small number but it is a real number.

2) it tends to zero

If we try the experiment an infinite number of times, does one of the possibilities (1) or 2)), neither or both give at least one time the result A with probability 1 (it means we are sure it happens at least once) ? (I guess the 1) works but I am not so good at prob.)

thanks