# Indexed Summation of Sum of Mappings

## Theorem

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $a, b$ be integers.

Let $\closedint a b$ denote the integer interval between $a$ and $b$.

Let $f, g: \closedint a b \to \mathbb A$ be mappings.

Let $h = f + g$ be their pointwise sum.

Then we have the equality of indexed summations:

$\displaystyle \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$

## Proof

The proof proceeds by induction on $b$.

For all $b \in \Z_{\ge 0}$, let $\map P b$ be the proposition:

$\displaystyle \sum_{i \mathop = a}^b \map h i = \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$

### Basis for the Induction

Let $b < a$.

Then all indexed summations are zero.

Because $0 = 0 + 0$, the result follows.

### Basis for the Induction

Let $b = a$.

 $\displaystyle \sum_{i \mathop = a}^b \map h i$ $=$ $\displaystyle \sum_{i \mathop = a}^a \map h i$ $\displaystyle$ $=$ $\displaystyle \map h a$ $\displaystyle$ $=$ $\displaystyle \map f a + \map g a$ Definition of Pointwise Addition $\displaystyle$ $=$ $\displaystyle \sum_{i \mathop = a}^a \map f i + \sum_{i \mathop = a}^a \map g i$ $\displaystyle$ $=$ $\displaystyle \sum_{i \mathop = a}^b \map f i + \sum_{i \mathop = a}^b \map g i$ as $a = b$

Thus $\map P a$ is seen to hold.

This is our basis for the induction.

### Induction Hypothesis

Now it needs to be shown that, if $\map P k$ is true, where $k \ge a$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

$\displaystyle \sum_{i \mathop = a}^k \map h i = \sum_{i \mathop = a}^k \map f i + \sum_{i \mathop = a}^k \map g i$

from which it is to be shown that:

$\displaystyle \sum_{i \mathop = a}^{k + 1} \map h i = \sum_{i \mathop = a}^{k + 1} \map f i + \sum_{i \mathop = a}^{k + 1} \map g i$

### Induction Step

We have:

 $\displaystyle \sum_{i \mathop = a}^{k + 1} \map h i$ $=$ $\displaystyle \sum_{i \mathop = a}^k \map h i + \map h {k + 1}$ Definition of Indexed Summation $\displaystyle$ $=$ $\displaystyle \sum_{i \mathop = a}^k \map h i + \map f {k + 1} + \map g {k + 1}$ Definition of Pointwise Addition $\displaystyle$ $=$ $\displaystyle \sum_{i \mathop = a}^k \map f i + \sum_{i \mathop = a}^k \map g i + \map f {k + 1} + \map g {k + 1}$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle \sum_{i \mathop = a}^k \map f i + \map f {k + 1} + \sum_{i \mathop = a}^k \map g i + \map g {k + 1}$ Arithmetic Addition is Commutative $\displaystyle$ $=$ $\displaystyle \sum_{i \mathop = a}^{k + 1} \map f i + \sum_{i \mathop = a}^{k + 1} \map g i$ Definition of Indexed Summation

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

$\blacksquare$