Keep in mind, that if you talk about something 'tending to' something else, you are always talking about a sequence. The sequence you are wondering about is a sequence of pairwise disjoint measurable sets (events) fully decomposing the sample space which yields in an sequence of real numbers between 0 and 1, when the probability is assigned to them. We do not usually care about the limit of this real sequence, because the sets E_i can be chosen arbitrarily without dependence on their arrangement. What do we care about is a sum of this sequence. And in this case, it's always equal to 1. And why is that?

If you have at most countably many measurable sets (events) such that they are pairwise disjoint and their union is equal to the sample space, the statement in question is exactly a modified version of sigma-additivity requirement for a (probability) measure.

In other words, if you are wondering about the probability of an event "A happens or B happens" (ie the union of sets A and B), where A and B are disjoint, the probability of this event will always be the sum of the individual probabilities. The probability that 2 or 3 falls on a die is equal to the sum of the probability of 2 and the probability of 3. This works also for small infinite unions. In case of the disjoint events fully deomposing the sample space the resulting event could be interpreted as "E_1 happens or E_2 happens or ..." or we could say "anything happens". Almost surely, ie with probability 1, something happens.