Sum of Even Integers is Even/Proof 1

Theorem

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

Proof

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.

$\blacksquare$