Question

1. Show that absolute value of r 12 years ago

Answer 1

Part 1:

abs(r) = { if r < 0, then -r, otherwise r }

r < s => if r < 0, then -r < s, multiply by the factor, -1, on both sides => r > -s
if r > 0, then r < s.

Combining these two inequalities, we see that -s < r < s.

Part 2:

We are given that s is a non-zero positive rational number.

r > -s is true for any positive s if r is zero or any positive rational. If r < 0, then it is possible to choose an r smaller than -s. So, let us only consider zero or positive rational numbers for r.

r < s is true for any positive s if r is zero, otherwise you can pick an s smaller than r, e.g., let r = q/p. Then, choose s = (q-1)/p.

Thus, the only r which satisfies the inequality for any s is r = 0.