A proof by contradiction has three things the marker looks for by name: the assumption, stated; the contradiction, stated; the conclusion, stated. Working that arrives at something false without saying so is a proof missing its last mark.
Inequalities: ≤ and < are not interchangeable, and from A > B and C > D nothing follows about A/C and B/D. Know the standard starting points — a square is non-negative, AM–GM — and say which one you are using.