Analysis I : Couchy Sequences...

I never can get myself to understand this... So I will state what I have memorised.

A sequence, (a_n) is Cauchy if, for all epsilon > 0, there is a number, N, which is a natural number where |a_n - a_m| <> N

So how is this useful? Well, for starters, all cauchy sequences are convergent. Need a proof?

Also, every convergent sequence is cauchy ^^, here is the proof:

I will later do this in LaTeX, so the formulas are more understandable desu~^^. For now, bye bee. The empty spaces are empty by the way.

Foundations: Principle of Algebra

This one is easy:

For all polynomial of power x, there exists x number of solutions. This includes repeated roots and complex roots desu~ ^^.

Foundations: Principle of Arithmatics

The principle of Arithmatics states that for all numbers greater than 1, the number can be divided by prime numbers and every number has a unique multiplication of prime numbers. I.e. if we were to show a number using prime factorisation, no two numbers will have the same prime factorisation desu ^^.