The Cauchy Sequence

So, I've finally decided on a title for this thing. I used this math concept as the backbone of my medical school personal statement (which I thought was kinda creative....but that's just me).

A sequence

x_1, x_2, x_3...

of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer N such that for all natural numbers m,n > N

|x_m - x_n| < ε.

[on a personal note, I used this definition in the rough draft...apparently my English professor thought it was a little overly-ambitious and told me to change it...and I thought I was just being smart...]

So basically the idea is that as we continue along a sequence of numbers...say, for instance 1/x from 1 to infinity...that set of numbers is considered Cauchy that as we continue along the sequence, the numbers converge on a single number. In this case we would have 1/1, 1/2, 1/3,..., 1/1000,..., 1/10000000000,...

This set of numbers is naturally converging to 0.

It's sad that I have to use such an explanation of this concept to get the idea across, but my argumentation is that, although I live a very "divergent" lifestyle (I have no clue what I want to be, no goals, I want to do basically everything I can), I realize that everything I do in my life is converging to some pretty specific goals:

1. Be a good person
2. Help people out
3. Try to be happy and non-regretting in the way that I live my life

As for where my life is headed, being in medical school, I thought that I would eventually be a physician somewhere doing something medical (whether it makes me happy or not). The good thing about life is that although it is a sequence of events that will eventually converge to something, we have the opportunity to be able to change the value it can converge to...and that is, in my opinion, a pretty optimistic way to look at life (a very rare thing for me...teehee).

So here is where it truly begins. The beginning of this "blog," this "sequence." Let's see if this converges to anything...