A sequence is defined as a function mapping natural numbers to a given set. In this case, I choose my set to be the set of events characterising one's life. There are 2 ways to do this - events could be segregated based on relative importance, or in chronological order.
Arjun and I agree that relative importance would not be a very good classification method, since any 1 event in your life cannot be given maximum importance. So we would like to classify in chronological order.
However, we must first prove that the set of events occuring in one's life is a countable set. The proof is clearly trivial. One must relate all the events that have occured in his or her life. Since they can be listed, hence it is a countable set(possibly countably infinite) and can be mapped to naturals.
In case they were not, any uncountable set can be mapped to reals. Hence, we could map it to reals such that all previously occured events converge to the next element in the sequence, which will increase as life moves forward. ie, all your past deeds lead to your future deeds.
The next natural question would be whether this sequence is convergent. In that case, distance between 2 events must be defined. We define distance between any 2 specified events as the time difference between their occurences.
Arjun now claims that this will remain a constant, and hence the series is not convergent, ie. things which we would consider significant in our life occur at constant intervals of time. He said the proof is trivial, but I could attempt to give an example. Anything extreme which occurs becomes normal in a very short duration of time, and hence any similar event will not be considered significant in the neighbourhood of that event. For example, if we became millionaires overnight, possibly the first major party would be a significant event. Sooner or later, all parties will become commonplace and hence will not be noticed.
Thus, life is indeed a divergent sequence, as for any sequence to be convergent, the distance between 2 successive events must decrease to 0. However, here, it is nearly a constant(definitely not continously decreasing), and hence the sequence is not convergent.
