Binomial Theorem/Integral Index

Theorem
Let $X$ be one of the set of numbers $\N, \Z, \Q, \R, \C$.

Let $x, y \in X$.

Then:
 * $\displaystyle \forall n \in \Z_{>0}: \left({x + y}\right)^n = \sum_{k \mathop = 0}^n {n \choose k} x^{n-k} y^k$

where $\displaystyle {n \choose k}$ is $n$ choose $k$.

Base Case
For $n = 0$ we have:


 * $\displaystyle \left({x+y}\right)^0 = 1 = {0 \choose 0} x^{0-0} y^0 = \sum_{k \mathop = 0}^0 {0 \choose k} x^{0-k} y^k$

Therefore the base case holds.

Inductive Hypothesis
This is our inductive hypothesis:


 * $\displaystyle \forall n \ge 1: \left({x+y}\right)^n = \sum_{k \mathop = 0}^n {n \choose k} x^{n-k} y^k$

Inductive Step
This is our inductive step:

The result follows by the Principle of Mathematical Induction.