>>12052102Let us first do the natural numbers, then show that it follows for negative integers too
Definition: An integer x is even iff there exists an integer y such that
0 is a natural number. 0 multiplied by 2 is 0, which is even.
Now assume it has been proven for the th natural number.
is even, therefore by definition there exists an integer x such that
Adding 2 to either side yields
factoring 2 yields
x+1 is an integer (follows from definition of integers in conjunction with peano axiom 6), thus n+1 is even by definition.
By induction this holds for all natural numbers.
Now consider the general case. Either x is positive, negative, or 0. We have already shown the theorem it for positive and 0, so assume x is negative.
Then x can be expressed as -y where y is some positive integer.
Then 2x = -2y, We have already shown that 2y must be even since 2y is positive. Thus 2x is even.
