User:Maelin/Sandbox

Sequences
We will assume that the natural numbers (1, 2, 3, ...), integers (..., -2, -1, 0, 1, 2, ...) and rational numbers have been developed and are understood. We are first going to develop a new, larger set of numbers called the real numbers. First, we need to consider sequences of rational numbers. Sequences are defined for arbitrary sets as follows:


 * Definition: A sequence in a set X is an assignment of the natural numbers to elements of X. So for each natural number 1, 2, etc., a sequence chooses exactly one object from within X. That is, if our sequence is (ai), then the "first element of the sequence" is an object from X, denoted a1. The "second element of the sequence" is another object from X, denoted a2. Note that it is allowable for a sequence to choose the same element of X many times.

Sequences can be expressed in several ways. Sometimes, if the pattern is obvious, we might just explicitly write out the first few terms, e.g. the sequence in the integers (ai) = (1, -1, 2, -2, 3, -3, ...). Other times, we might give a rule by which we can calculate each particular term, e.g. the sequence in the natural numbers given by ai = 2i + 3, which gives the sequence (ai) = (5, 7, 9, 11, ...). Our rule can even, if we have express each term in terms of previous terms, as long as it gives the first few terms explicitly. Here are some examples of sequences in the natural numbers:


 * ai = i, gives the natural numbers themselves, as a sequence (ai) = (1, 2, 3, ...)
 * $$ \textstyle a_i = \sum_{j=1}^i i = \frac{i(i+1)}{2} $$ gives the sequence (a'sub>i ) = (1, 3, 6, 10, 15, 21, ...)
 * a1 = 1, a2 = 1, ai+2 = ai+1 + ai gives the Fibonacci sequence (a'sub>i ) = (1, 1, 2, 3, 5, 8, 13, ...)

Note that, in a set with more than two elements, there are lots of sequences that we can't describe in a practical way at all. Fortunately, we will not need to use any of these 'awkward' sequences explicitly.

Now we will consider convergence, but first, we need to quickly examine the idea of a metric space. A metric space is simply a set that comes with a way of measuring the distance between its elements. So, for example, our three important number sets, the naturals, the integers, and the rationals, all have a way of finding the distance between their elements using this formula: if a, b are in the set, then d(a, b) =

$$ \begin{align} \int_{-\infty}^\infty f(x) dx & = \sum_{i=-\infty}^\infty \int_i^i f(x) dx \\ & = \lim_{\begin{smallmatrix}m \to -\infty \\ n \to \infty \end{smallmatrix}} \sum_{i=m}^n \left( \lim_{\begin{smallmatrix}a \to i^+ \\ b \to (i+1)^- \end{smallmatrix}} \int_a^b f(x) dx \right) \\ & = \lim_{\begin{smallmatrix}m \to -\infty \\ n \to \infty \end{smallmatrix}} \sum_{i=m}^n \left( \lim_{\begin{smallmatrix}a \to i^+ \\ b \to (i+1)^- \end{smallmatrix}} F(b) - F(a) \right) \end{align}$$