Proving that a finite set is a convex set

Question

Assignment:

Q1. Prove that if S is a finite set then (S) is a convex and compact set.

Q2. Let A ⊆ Rⁿ and B ⊆ R^m be two compact sets. Prove that the product set A × B ⊆ R^n+m is a compact set.

Q3. Let A ⊆ Rⁿ and B ⊆ R^m be two convex sets. Prove that the product set A × B ⊆ R^n+m is a convex set.

Provide complete and step by step solution for the question and show calculations and use formulas.