# Sum of Even Integers is Even

## Theorem

The sum of any finite number of even integers is itself even.

## Proof 1

Proof by induction:

For all $n \in \N$, let $\map P n$ be the proposition:

- The sum of $n$ even integers is an even integer.

$\map P 1$ is trivially true, as this just says:

- The sum of $1$ even integer is an even integer.

The sum of $0$ even integers is understood, from the definition of a vacuous summation, to be $0$, which is even.

So $\map P 0$ is also true.

### Basis for the Induction

$\map P 2$ is the case:

- The sum of any two even integers is itself even.

Consider two even integers $x$ and $y$.

Since they are even, they can be written as $x = 2 a$ and $y = 2 b$ respectively for integers $a$ and $b$.

Therefore, the sum is:

- $x + y = 2 a + 2 b = 2 \paren {a + b}$

Thus $x + y$ has $2$ as a divisor and is even by definition.

This is our basis for the induction.

$\Box$

### Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

- The sum of any $k$ even integers is itself even.

Then we need to show:

- The sum of any $k+1$ even integers is itself even.

### Induction Step

This is our induction step:

Consider the sum of any $k + 1$ even integers.

This is the sum of:

- $k$ even integers (which is even by the induction hypothesis)

and:

- another even integer.

That is, it is the sum of two even integers.

By the basis for the induction, this is also even.

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

That is:

- The sum of any finite number of even integers is itself even.

$\blacksquare$

## Proof 2

Let $S = \set {r_1, r_2, \ldots, r_n}$ be a set of $n$ even numbers.

By definition of even number, this can be expressed as:

- $S = \set {2 s_1, 2 s_2, \ldots, 2 s_n}$

where:

- $\forall k \in \closedint 1 n: r_k = 2 s_k$

Then:

\(\ds \sum_{k \mathop = 1}^n r_k\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n 2 s_k\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 \sum_{k \mathop = 1}^n s_k\) |

Thus, by definition, $\ds \sum_{k \mathop = 1}^n r_k$ is even.

$\blacksquare$