Question

1)prove that if x is rational and x not equal to 0, then 1/x is rational.

2) prove that there is a positive integers that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?


For 1) use the definition of a non-zero rational number.
Defn: A non-zero rational is any number that can be expressed as p/q where p and q are non-zero integers.
If p/q is rational, then how about q/p?

For 2) you should be able to supply both a constructive and non-consructive proof.
Since the sum of any finite set of integers is an integer, there exists an integer for the sum of the first +n integers.
There is a formula for this, but I'll let you work on this.

Answer 1

Constructive proof:
Let n be a positive integer. Then the sum of the first n positive integers is equal to n(n+1)/2. Therefore, for any positive integer n, there exists a positive integer that equals the sum of the positive integers not exceeding it.

Nonconstructive proof:
Let n be a positive integer. By the Well-Ordering Principle, there exists a positive integer that equals the sum of the positive integers not exceeding it.