Prove that if s is a finite set then s is a convex and

Question

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

2. 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.

3. 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.