Completeness (Intuition) Math Example 2

Solution

1. 1
   For completeness, induction requires two parts: a base case and an inductive step covering all remaining cases.
2. 2
   Base case ($n=1$): $1^2 = 1 \ge 1$. True.
3. 3
   Inductive step: Assume $k^2 \ge k$ for some $k \ge 1$. Show $(k+1)^2 \ge k+1$.
4. 4
   $(k+1)^2 = k^2+2k+1 \ge k+2k+1 = 3k+1 \ge k+1$ (since $2k \ge 0$). True.
5. 5
   Both parts are covered — the proof is complete for all $n \ge 1$.