Analytical Patterns

Abstract

Analysis is introduced. Sequences are defined. The triangle inequality is presented. The definition of the limit of a sequence is given. Basic properties of limits are established and simple counterexamples presented. The poly-epsilon pattern is presented as a way of proving simple results in analysis. Infinite series are introduced and their convergence as a limit of partial sums is defined. The convergence of sums of powers of $n$ is examined. The concept of a continuous function is defined in terms of the convergence of functions of limits of sequences. Examples of continuous functions are given. The Intermediate Value property is proven for continuous functions. The attainment of maxima and minima on finite closed intervals is also shown. These results are applied to prove that every complex polynomial always has a zero.