Let a b c disin r if a gt 0 and b gt 0 then ab gt 0 if a gt

Question

1. Let a, b, c, d∈ R. If a > 0 and b > 0, then ab > 0. If a > 0 and b ≥ 0, then ab ≥ 0. If a ≥0 and b ≥0, then ab ≥ 0. Need to prove this.

2. Let q ∈ R and x ∈ R - Q. Prove that q + x ∈ R - Q.

3. Prove that ½ + ¼ + 1/8 + ? + 1/2ⁿ =1 - 1/2ⁿ for all n ∈N.